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Particular values of the Riemann zeta function
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<h2 id="the-riemann-zeta-function-at-0-and-1">The Riemann zeta function at 0 and 1</h2> <p>At <a href="/facts/Zero_(complex_analysis)/gnA3U8VP">zero</a>, one has ζ ( 0 ) = B 1 − = − B 1 + = − 1 2 {\displaystyle \zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!} </p><p>At 1 there is a <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">pole</a>, so <i>ζ</i>(1) is not finite but the left and right limits are: lim ε → 0 ± ζ ( 1 + ε ) = ± ∞ {\displaystyle \lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty } Since it is a pole of first order, it has a <a href="/facts/Residue_(complex_analysis)/GllQJk32">complex residue</a> lim ε → 0 ε ζ ( 1 + ε ) = 1 . {\displaystyle \lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.} </p> <h2 id="positive-integers">Positive integers</h2> <h3>Even positive integers</h3> <p>For the even positive integers n {\displaystyle n} , one has the relationship to the <a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a> B n {\displaystyle B_{n}} : </p><p> ζ ( n ) = ( − 1 ) n 2 + 1 ( 2 π ) n B n 2 ( n ! ) . {\displaystyle \zeta (n)=(-1)^{{\tfrac {n}{2}}+1}{\frac {(2\pi )^{n}B_{n}}{2(n!)}}\,.} </p><p>The computation of ζ ( 2 ) {\displaystyle \zeta (2)} is known as the <a href="/facts/Basel_problem/2PManmlC">Basel problem</a>. The value of ζ ( 4 ) {\displaystyle \zeta (4)} is related to the <a href="/facts/Stefan%25E2%2580%2593Boltzmann_law/vA4tbwKC">Stefan–Boltzmann law</a> and <a href="/facts/Wien_approximation/9630y3IE">Wien approximation</a> in physics. The first few values are given by: ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + ⋯ = π 2 6 ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + ⋯ = π 4 90 ζ ( 6 ) = 1 + 1 2 6 + 1 3 6 + ⋯ = π 6 945 ζ ( 8 ) = 1 + 1 2 8 + 1 3 8 + ⋯ = π 8 9450 ζ ( 10 ) = 1 + 1 2 10 + 1 3 10 + ⋯ = π 10 93555 ζ ( 12 ) = 1 + 1 2 12 + 1 3 12 + ⋯ = 691 π 12 638512875 ζ ( 14 ) = 1 + 1 2 14 + 1 3 14 + ⋯ = 2 π 14 18243225 ζ ( 16 ) = 1 + 1 2 16 + 1 3 16 + ⋯ = 3617 π 16 325641566250 . {\displaystyle {\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\\[4pt]\zeta (16)&=1+{\frac {1}{2^{16}}}+{\frac {1}{3^{16}}}+\cdots ={\frac {3617\pi ^{16}}{325641566250}}\,.\end{aligned}}} </p><p>Taking the limit n → ∞ {\displaystyle n\rightarrow \infty } , one obtains ζ ( ∞ ) = 1 {\displaystyle \zeta (\infty )=1} . </p> Selected values for even integers<table><tbody><tr><th scope="col">Value</th><th scope="col">Decimal expansion</th><th scope="col">Source</th></tr><tr><td> ζ ( 2 ) {\displaystyle \zeta (2)} </td><td>1.6449340668482264364...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013661</td></tr><tr><td> ζ ( 4 ) {\displaystyle \zeta (4)} </td><td>1.0823232337111381915...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013662</td></tr><tr><td> ζ ( 6 ) {\displaystyle \zeta (6)} </td><td>1.0173430619844491397...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013664</td></tr><tr><td> ζ ( 8 ) {\displaystyle \zeta (8)} </td><td>1.0040773561979443393...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013666</td></tr><tr><td> ζ ( 10 ) {\displaystyle \zeta (10)} </td><td>1.0009945751278180853...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013668</td></tr><tr><td> ζ ( 12 ) {\displaystyle \zeta (12)} </td><td>1.0002460865533080482...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013670</td></tr><tr><td> ζ ( 14 ) {\displaystyle \zeta (14)} </td><td>1.0000612481350587048...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013672</td></tr><tr><td> ζ ( 16 ) {\displaystyle \zeta (16)} </td><td>1.0000152822594086518...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013674</td></tr></tbody></table> <p>The relationship between zeta at the positive even integers and powers of pi may be written as </p><p> a n ζ ( 2 n ) = π 2 n b n {\displaystyle a_{n}\zeta (2n)=\pi ^{2n}b_{n}} </p><p>where a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} are coprime positive integers for all n {\displaystyle n} . These are given by the integer sequences <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A002432 and <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A046988, respectively, in <a href="/facts/OEIS/yow6cVsU">OEIS</a>. Some of these values are reproduced below: </p> coefficients<table><tbody><tr><th><i>n</i></th><th><i>a</i><i>n</i></th><th><i>b</i><i>n</i></th></tr><tr><td>1</td><td>6</td><td>1</td></tr><tr><td>2</td><td>90</td><td>1</td></tr><tr><td>3</td><td>945</td><td>1</td></tr><tr><td>4</td><td>9450</td><td>1</td></tr><tr><td>5</td><td>93555</td><td>1</td></tr><tr><td>6</td><td>638512875</td><td>691</td></tr><tr><td>7</td><td>18243225</td><td>2</td></tr><tr><td>8</td><td>325641566250</td><td>3617</td></tr><tr><td>9</td><td>38979295480125</td><td>43867</td></tr><tr><td>10</td><td>1531329465290625</td><td>174611</td></tr><tr><td>11</td><td>13447856940643125</td><td>155366</td></tr><tr><td>12</td><td>201919571963756521875</td><td>236364091</td></tr><tr><td>13</td><td>11094481976030578125</td><td>1315862</td></tr><tr><td>14</td><td>564653660170076273671875</td><td>6785560294</td></tr><tr><td>15</td><td>5660878804669082674070015625</td><td>6892673020804</td></tr><tr><td>16</td><td>62490220571022341207266406250</td><td>7709321041217</td></tr><tr><td>17</td><td>12130454581433748587292890625</td><td>151628697551</td></tr></tbody></table> <p>If we let η n = b n / a n {\displaystyle \eta _{n}=b_{n}/a_{n}} be the coefficient of π 2 n {\displaystyle \pi ^{2n}} as above, ζ ( 2 n ) = ∑ ℓ = 1 ∞ 1 ℓ 2 n = η n π 2 n {\displaystyle \zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}} then we find recursively, </p><p> η 1 = 1 / 6 η n = ∑ ℓ = 1 n − 1 ( − 1 ) ℓ − 1 η n − ℓ ( 2 ℓ + 1 ) ! + ( − 1 ) n + 1 n ( 2 n + 1 ) ! {\displaystyle {\begin{aligned}\eta _{1}&=1/6\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}} </p><p>This recurrence relation may be derived from that for the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a>. </p><p>Also, there is another recurrence: </p><p> ζ ( 2 n ) = 1 n + 1 2 ∑ k = 1 n − 1 ζ ( 2 k ) ζ ( 2 n − 2 k ) for n > 1 {\displaystyle \zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{k=1}^{n-1}\zeta (2k)\zeta (2n-2k)\quad {\text{ for }}\quad n>1} which can be proved, using that d d x cot ( x ) = − 1 − cot 2 ( x ) {\displaystyle {\frac {d}{dx}}\cot(x)=-1-\cot ^{2}(x)} </p><p>The values of the zeta function at non-negative even integers have the <a href="/facts/Generating_function/K7UVjnjL">generating function</a>: ∑ n = 0 ∞ ζ ( 2 n ) x 2 n = − π x 2 cot ( π x ) = − 1 2 + π 2 6 x 2 + π 4 90 x 4 + π 6 945 x 6 + ⋯ {\displaystyle \sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots } Since lim n → ∞ ζ ( 2 n ) = 1 {\displaystyle \lim _{n\rightarrow \infty }\zeta (2n)=1} The formula also shows that for n ∈ N , n → ∞ {\displaystyle n\in \mathbb {N} ,n\rightarrow \infty } , | B 2 n | ∼ ( 2 n ) ! 2 ( 2 π ) 2 n {\displaystyle \left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}} </p> <h3>Odd positive integers</h3> <p>The sum of the <a href="/facts/Harmonic_series_(mathematics)/8KGDHKTg">harmonic series</a> is infinite. ζ ( 1 ) = 1 + 1 2 + 1 3 + ⋯ = ∞ {\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!} </p><p>The value <i>ζ</i>(3) is also known as <a href="/facts/Ap%25C3%25A9ry%2527s_constant/59do2TbQ">Apéry's constant</a> and has a role in the electron's gyromagnetic ratio. The value <i>ζ</i>(3) also appears in <a href="/facts/Planck%2527s_law/7Umkdb9m">Planck's law</a>. These and additional values are: </p> Selected values for odd integers<table><tbody><tr><th scope="col">Value</th><th scope="col">Decimal expansion</th><th scope="col">Source</th></tr><tr><td><a href="/facts/Ap%25C3%25A9ry%2527s_constant/59do2TbQ"> ζ ( 3 ) {\displaystyle \zeta (3)} </a></td><td>1.2020569031595942853...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A002117</td></tr><tr><td> ζ ( 5 ) {\displaystyle \zeta (5)} </td><td>1.0369277551433699263...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013663</td></tr><tr><td> ζ ( 7 ) {\displaystyle \zeta (7)} </td><td>1.0083492773819228268...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013665</td></tr><tr><td> ζ ( 9 ) {\displaystyle \zeta (9)} </td><td>1.0020083928260822144...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013667</td></tr><tr><td> ζ ( 11 ) {\displaystyle \zeta (11)} </td><td>1.0004941886041194645...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013669</td></tr><tr><td> ζ ( 13 ) {\displaystyle \zeta (13)} </td><td>1.0001227133475784891...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013671</td></tr><tr><td> ζ ( 15 ) {\displaystyle \zeta (15)} </td><td>1.0000305882363070204...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013673</td></tr></tbody></table> <p>It is known that <i>ζ</i>(3) is irrational (<a href="/facts/Ap%25C3%25A9ry%2527s_theorem/lY9DCz1b">Apéry's theorem</a>) and that infinitely many of the numbers <i>ζ</i>(2<i>n</i> + 1) : <i>n</i> ∈ N {\displaystyle \mathbb {N} } , are irrational.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of <i>ζ</i>(5), <i>ζ</i>(7), <i>ζ</i>(9), or <i>ζ</i>(11) is irrational.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>The positive odd integers of the zeta function appear in physics, specifically <a href="/facts/Correlation_function_(statistical_mechanics)/eWP84mzu">correlation functions</a> of antiferromagnetic <a href="/facts/Heisenberg_model_(quantum)/8RPu8n5n">XXX spin chain</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Most of the identities following below are provided by <a href="/facts/Simon_Plouffe/JeISXKeT">Simon Plouffe</a>. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations. </p><p>Plouffe stated the following identities without proof.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Proofs were later given by other authors.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h4><i>ζ</i>(5)</h4> <p> ζ ( 5 ) = 1 294 π 5 − 72 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) − 2 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 ) ζ ( 5 ) = 12 ∑ n = 1 ∞ 1 n 5 sinh ( π n ) − 39 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) + 1 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 ) {\displaystyle {\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}+{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}} </p> <h4><i>ζ</i>(7)</h4> <p> ζ ( 7 ) = 19 56700 π 7 − 2 ∑ n = 1 ∞ 1 n 7 ( e 2 π n − 1 ) {\displaystyle \zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!} </p><p>Note that the sum is in the form of a <a href="/facts/Lambert_series/HFU5HfCR">Lambert series</a>. </p> <h4><i>ζ</i>(2<i>n</i> + 1)</h4> <p>By defining the quantities </p><p> S ± ( s ) = ∑ n = 1 ∞ 1 n s ( e 2 π n ± 1 ) {\displaystyle S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}} </p><p>a series of relationships can be given in the form </p><p> 0 = a n ζ ( n ) − b n π n + c n S − ( n ) + d n S + ( n ) {\displaystyle 0=a_{n}\zeta (n)-b_{n}\pi ^{n}+c_{n}S_{-}(n)+d_{n}S_{+}(n)} </p><p>where <i>a</i><i>n</i>, <i>b</i><i>n</i>, <i>c</i><i>n</i> and <i>d</i><i>n</i> are positive integers. Plouffe gives a table of values: </p> coefficients<table><tbody><tr><th><i>n</i></th><th><i>a</i><i>n</i></th><th><i>b</i><i>n</i></th><th><i>c</i><i>n</i></th><th><i>d</i><i>n</i></th></tr><tr><td>3</td><td>180</td><td>7</td><td>360</td><td>0</td></tr><tr><td>5</td><td>1470</td><td>5</td><td>3024</td><td>84</td></tr><tr><td>7</td><td>56700</td><td>19</td><td>113400</td><td>0</td></tr><tr><td>9</td><td>18523890</td><td>625</td><td>37122624</td><td>74844</td></tr><tr><td>11</td><td>425675250</td><td>1453</td><td>851350500</td><td>0</td></tr><tr><td>13</td><td>257432175</td><td>89</td><td>514926720</td><td>62370</td></tr><tr><td>15</td><td>390769879500</td><td>13687</td><td>781539759000</td><td>0</td></tr><tr><td>17</td><td>1904417007743250</td><td>6758333</td><td>3808863131673600</td><td>29116187100</td></tr><tr><td>19</td><td>21438612514068750</td><td>7708537</td><td>42877225028137500</td><td>0</td></tr><tr><td>21</td><td>1881063815762259253125</td><td>68529640373</td><td>3762129424572110592000</td><td>1793047592085750</td></tr></tbody></table> <p>These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below. </p><p>A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="negative-integers">Negative integers</h2> <p>In general, for negative integers (and also zero), one has </p><p> ζ ( − n ) = ( − 1 ) n B n + 1 n + 1 {\displaystyle \zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}} </p><p>The so-called "trivial zeros" occur at the negative even integers: </p><p> ζ ( − 2 n ) = 0 {\displaystyle \zeta (-2n)=0} (<a href="/facts/Ramanujan_summation/3GNBWCoB">Ramanujan summation</a>) </p><p>The first few values for negative odd integers are </p><p> ζ ( − 1 ) = − 1 12 ζ ( − 3 ) = 1 120 ζ ( − 5 ) = − 1 252 ζ ( − 7 ) = 1 240 ζ ( − 9 ) = − 1 132 ζ ( − 11 ) = 691 32760 ζ ( − 13 ) = − 1 12 {\displaystyle {\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}} </p><p>However, just like the <a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a>, these do not stay small for increasingly negative odd values. For details on the first value, see <a href="/facts/1_%252B_2_%252B_3_%252B_4_%252B_%25C2%25B7_%25C2%25B7_%25C2%25B7/6qFgKG0Y">1 + 2 + 3 + 4 + · · ·</a>. </p><p>So <i>ζ</i>(<i>m</i>) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers. </p> <h2 id="derivatives">Derivatives</h2> <p>The derivative of the zeta function at the negative even integers is given by </p><p> ζ ′ ( − 2 n ) = ( − 1 ) n ( 2 n ) ! 2 ( 2 π ) 2 n ζ ( 2 n + 1 ) . {\displaystyle \zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1)\,.} </p><p>The first few values of which are </p><p> ζ ′ ( − 2 ) = − ζ ( 3 ) 4 π 2 ζ ′ ( − 4 ) = 3 4 π 4 ζ ( 5 ) ζ ′ ( − 6 ) = − 45 8 π 6 ζ ( 7 ) ζ ′ ( − 8 ) = 315 4 π 8 ζ ( 9 ) . {\displaystyle {\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}} </p><p>One also has </p><p> ζ ′ ( 0 ) = − 1 2 ln ( 2 π ) ζ ′ ( − 1 ) = 1 12 − ln A ζ ′ ( 2 ) = 1 6 π 2 ( γ + ln 2 − 12 ln A + ln π ) {\displaystyle {\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}} </p><p>where <i>A</i> is the <a href="/facts/Glaisher%25E2%2580%2593Kinkelin_constant/lK8noN4A">Glaisher–Kinkelin constant</a>. The first of these identities implies that the regularized product of the reciprocals of the positive integers is 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} , thus the amusing "equation" ∞ ! = 2 π {\displaystyle \infty !={\sqrt {2\pi }}} .<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>From the logarithmic derivative of the functional equation, </p><p> 2 ζ ′ ( 1 / 2 ) ζ ( 1 / 2 ) = log ( 2 π ) + π cos ( π / 4 ) 2 sin ( π / 4 ) − Γ ′ ( 1 / 2 ) Γ ( 1 / 2 ) = log ( 2 π ) + π 2 + 2 log 2 + γ . {\displaystyle 2{\frac {\zeta '(1/2)}{\zeta (1/2)}}=\log(2\pi )+{\frac {\pi \cos(\pi /4)}{2\sin(\pi /4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.} </p> Selected derivatives<table><tbody><tr><th scope="col">Value</th><th scope="col">Decimal expansion</th><th scope="col">Source</th></tr><tr><td> ζ ′ ( 3 ) {\displaystyle \zeta '(3)} </td><td>−0.19812624288563685333...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A244115</td></tr><tr><td> ζ ′ ( 2 ) {\displaystyle \zeta '(2)} </td><td>−0.93754825431584375370...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A073002</td></tr><tr><td> ζ ′ ( 0 ) {\displaystyle \zeta '(0)} </td><td>−0.91893853320467274178...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A075700</td></tr><tr><td> ζ ′ ( − 1 2 ) {\displaystyle \zeta '(-{\tfrac {1}{2}})} </td><td>−0.36085433959994760734...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A271854</td></tr><tr><td> ζ ′ ( − 1 ) {\displaystyle \zeta '(-1)} </td><td>−0.16542114370045092921...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A084448</td></tr><tr><td> ζ ′ ( − 2 ) {\displaystyle \zeta '(-2)} </td><td>−0.030448457058393270780...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A240966</td></tr><tr><td> ζ ′ ( − 3 ) {\displaystyle \zeta '(-3)} </td><td>+0.0053785763577743011444...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259068</td></tr><tr><td> ζ ′ ( − 4 ) {\displaystyle \zeta '(-4)} </td><td>+0.0079838114502686242806...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259069</td></tr><tr><td> ζ ′ ( − 5 ) {\displaystyle \zeta '(-5)} </td><td>−0.00057298598019863520499...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259070</td></tr><tr><td> ζ ′ ( − 6 ) {\displaystyle \zeta '(-6)} </td><td>−0.0058997591435159374506...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259071</td></tr><tr><td> ζ ′ ( − 7 ) {\displaystyle \zeta '(-7)} </td><td>−0.00072864268015924065246...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259072</td></tr><tr><td> ζ ′ ( − 8 ) {\displaystyle \zeta '(-8)} </td><td>+0.0083161619856022473595...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A259073</td></tr></tbody></table> <h2 id="series-involving-n">Series involving <i>ζ</i>(<i>n</i>)</h2> <p>The following sums can be derived from the generating function: ∑ k = 2 ∞ ζ ( k ) x k − 1 = − ψ 0 ( 1 − x ) − γ {\displaystyle \sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma } where <i>ψ</i>0 is the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>. </p><p> ∑ k = 2 ∞ ( ζ ( k ) − 1 ) = 1 ∑ k = 1 ∞ ( ζ ( 2 k ) − 1 ) = 3 4 ∑ k = 1 ∞ ( ζ ( 2 k + 1 ) − 1 ) = 1 4 ∑ k = 2 ∞ ( − 1 ) k ( ζ ( k ) − 1 ) = 1 2 {\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}} </p><p>Series related to the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a> (denoted by <i>γ</i>) are ∑ k = 2 ∞ ( − 1 ) k ζ ( k ) k = γ ∑ k = 2 ∞ ζ ( k ) − 1 k = 1 − γ ∑ k = 2 ∞ ( − 1 ) k ζ ( k ) − 1 k = ln 2 + γ − 1 {\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}} </p><p>and using the principal value ζ ( k ) = lim ε → 0 ζ ( k + ε ) + ζ ( k − ε ) 2 {\displaystyle \zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}}} which of course affects only the value at 1, these formulae can be stated as </p><p> ∑ k = 1 ∞ ( − 1 ) k ζ ( k ) k = 0 ∑ k = 1 ∞ ζ ( k ) − 1 k = 0 ∑ k = 1 ∞ ( − 1 ) k ζ ( k ) − 1 k = ln 2 {\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}} </p><p>and show that they depend on the principal value of <i>ζ</i>(1) = <i>γ</i> . </p> <h2 id="nontrivial-zeros">Nontrivial zeros</h2> <p class="note">Main article: <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a></p> <p>Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a> states that the real part of every nontrivial zero must be 1/2. In other words, all known nontrivial zeros of the Riemann zeta are of the form <i>z</i> = 1/2 + <i>y</i>i where <i>y</i> is a real number. The following table contains the decimal expansion of Im(<i>z</i>) for the first few nontrivial zeros: </p> Selected nontrivial zeros<table><tbody><tr><th scope="col">Decimal expansion of Im(<i>z</i>)</th><th scope="col">Source</th></tr><tr><td>14.134725141734693790...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A058303</td></tr><tr><td>21.022039638771554992...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065434</td></tr><tr><td>25.010857580145688763...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065452</td></tr><tr><td>30.424876125859513210...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065453</td></tr><tr><td>32.935061587739189690...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A192492</td></tr><tr><td>37.586178158825671257...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A305741</td></tr><tr><td>40.918719012147495187...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A305742</td></tr><tr><td>43.327073280914999519...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A305743</td></tr><tr><td>48.005150881167159727...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A305744</td></tr><tr><td>49.773832477672302181...</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A306004</td></tr></tbody></table> <p><a href="/facts/Andrew_Odlyzko/rBsYtPz5">Andrew Odlyzko</a> computed the first 2 million nontrivial zeros accurate to within 4×10−9, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> A table of about 103 billion zeros with high precision (of ±2−102≈±2·10−31) is available for interactive access and download (although in a very inconvenient compressed format) via LMFDB.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="ratios">Ratios</h2> <p>Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting <a href="/facts/Particular_values_of_the_gamma_function/5yPGmC7b">particular values of the gamma function</a> into the functional equation </p><p> ζ ( s ) = 2 s π s − 1 sin ( π s 2 ) Γ ( 1 − s ) ζ ( 1 − s ) {\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)} </p><p>We have simple relations for half-integer arguments </p><p> ζ ( 3 / 2 ) ζ ( − 1 / 2 ) = − 4 π ζ ( 5 / 2 ) ζ ( − 3 / 2 ) = − 16 π 2 3 ζ ( 7 / 2 ) ζ ( − 5 / 2 ) = 64 π 3 15 ζ ( 9 / 2 ) ζ ( − 7 / 2 ) = 256 π 4 105 {\displaystyle {\begin{aligned}{\frac {\zeta (3/2)}{\zeta (-1/2)}}&=-4\pi \\{\frac {\zeta (5/2)}{\zeta (-3/2)}}&=-{\frac {16\pi ^{2}}{3}}\\{\frac {\zeta (7/2)}{\zeta (-5/2)}}&={\frac {64\pi ^{3}}{15}}\\{\frac {\zeta (9/2)}{\zeta (-7/2)}}&={\frac {256\pi ^{4}}{105}}\end{aligned}}} </p><p>Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation </p><p> Γ ( 3 4 ) = ( π 2 ) 1 4 AGM ( 2 , 1 ) 1 2 {\displaystyle \Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}} </p><p>is the zeta ratio relation </p><p> ζ ( 3 / 4 ) ζ ( 1 / 4 ) = 2 π ( 2 − 2 ) AGM ( 2 , 1 ) {\displaystyle {\frac {\zeta (3/4)}{\zeta (1/4)}}=2{\sqrt {\frac {\pi }{(2-{\sqrt {2}})\operatorname {AGM} \left({\sqrt {2}},1\right)}}}} </p><p>where AGM is the <a href="/facts/Arithmetic%25E2%2580%2593geometric_mean/8PRN9UB4">arithmetic–geometric mean</a>. In a similar vein, it is possible to form radical relations, such as from </p> Γ ( 1 5 ) 2 Γ ( 1 10 ) Γ ( 3 10 ) = 1 + 5 2 7 10 5 4 {\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}} <p>the analogous zeta relation is </p><p> ζ ( 1 / 5 ) 2 ζ ( 7 / 10 ) ζ ( 9 / 10 ) ζ ( 1 / 10 ) ζ ( 3 / 10 ) ζ ( 4 / 5 ) 2 = ( 5 − 5 ) ( 10 + 5 + 5 ) 10 ⋅ 2 3 10 {\displaystyle {\frac {\zeta (1/5)^{2}\zeta (7/10)\zeta (9/10)}{\zeta (1/10)\zeta (3/10)\zeta (4/5)^{2}}}={\frac {(5-{\sqrt {5}})\left({\sqrt {10}}+{\sqrt {5+{\sqrt {5}}}}\right)}{10\cdot 2^{\tfrac {3}{10}}}}} </p> <h2 id="further-reading">Further reading</h2> <ul><li>Ciaurri, Óscar; Navas, Luis M.; Ruiz, Francisco J.; Varona, Juan L. (May 2015). "A Simple Computation of <i>ζ</i>(2<i>k</i>)". <i>The American Mathematical Monthly</i>. 122 (5): 444–451. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1209.5030">1209.5030</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4169%2Famer.math.monthly.122.5.444">10.4169/amer.math.monthly.122.5.444</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.444">10.4169/amer.math.monthly.122.5.444</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:207521195">207521195</a>.</li> <li><a href="/facts/Simon_Plouffe/JeISXKeT">Simon Plouffe</a>, "<a href="http://www.lacim.uqam.ca/~plouffe/identities.html">Identities inspired from Ramanujan Notebooks</a> <a href="https://web.archive.org/web/20090130142844/http://www.lacim.uqam.ca/~plouffe/identities.html">Archived</a> 2009-01-30 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>", (1998).</li> <li><a href="/facts/Simon_Plouffe/JeISXKeT">Simon Plouffe</a>, "<a href="http://www.lacim.uqam.ca/~plouffe/inspired22.html">Identities inspired by Ramanujan Notebooks part 2</a> <a href="http://www.lacim.uqam.ca/~plouffe/inspired2.pdf">PDF</a> <a href="https://web.archive.org/web/20110926231624/http://www.lacim.uqam.ca/~plouffe/inspired2.pdf">Archived</a> 2011-09-26 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>" (2006).</li> <li>Vepstas, Linas (2006). <a href="http://www.linas.org/math/plouffe-ram.pdf">"On Plouffe's Ramanujan identities"</a> (PDF). <i>The Ramanujan Journal</i>. 27 (3): 387–408. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.NT/0609775">math.NT/0609775</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11139-011-9335-9">10.1007/s11139-011-9335-9</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:8789411">8789411</a>.</li> <li><a href="/facts/Wadim_Zudilin/jU86aGjY">Zudilin, Wadim</a> (2001). "One of the Numbers <i>ζ</i>(5), <i>ζ</i>(7), <i>ζ</i>(9), <i>ζ</i>(11) Is Irrational". <i><a href="/facts/Russian_Mathematical_Surveys/g3lszK89">Russian Mathematical Surveys</a></i>. 56 (4): 774–776. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2001RuMaS..56..774Z">2001RuMaS..56..774Z</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1070%2FRM2001v056n04ABEH000427">10.1070/RM2001v056n04ABEH000427</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1861452">1861452</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250734661">250734661</a>. <a href="https://wain.mi-ras.ru/PS/zeta5-11$.pdf">PDF</a> <a href="https://wain.mi-ras.ru/PS/zeta5-11.pdf">PDF Russian</a> <a href="https://wain.mi-ras.ru/PS/zeta5-11.ps.gz">PS Russian</a></li> <li>Nontrival zeros reference by <a href="/facts/Andrew_Odlyzko/rBsYtPz5">Andrew Odlyzko</a>: <ul><li><a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">Bibliography</a></li> <li><a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html">Tables</a></li></ul></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270. arXiv:math/0008051. Bibcode:2000CRASM.331..267R. doi:10.1016/S0764-4442(00)01624-4. S2CID 119678120. <a href="/wiki/Tanguy_Rivoal" target="_blank">/wiki/Tanguy_Rivoal</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/rm2001v056n04abeh000427. S2CID 250734661. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A. 35 (20): 4443–4452. arXiv:cond-mat/0202346. Bibcode:2002JPhA...35.4443B. doi:10.1088/0305-4470/35/20/305. S2CID 119143600.. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Identities for Zeta(2*n+1)". <a href="http://www.plouffe.fr/simon/identities.html" target="_blank">http://www.plouffe.fr/simon/identities.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Formulas for Odd Zeta Values and Powers of Pi". <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Lopatto/lopatto2.html" target="_blank">https://cs.uwaterloo.ca/journals/JIS/VOL14/Lopatto/lopatto2.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927. <a href="https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=294&option_lang=eng" target="_blank">https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=294&option_lang=eng</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996). <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993). <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Muñoz García, E.; Pérez Marco, R. (2008), "The Product Over All Primes is 4 π 2 {\displaystyle 4\pi ^{2}} ", Commun. Math. Phys. (277): 69–81. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Odlyzko, Andrew. "Tables of zeros of the Riemann zeta function". Retrieved 7 September 2022. <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html" target="_blank">http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Odlyzko, Andrew. "Papers on Zeros of the Riemann Zeta Function and Related Topics". Retrieved 7 September 2022. <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html" target="_blank">http://www.dtc.umn.edu/~odlyzko/doc/zeta.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>LMFDB: Zeros of ζ(s) <a href="https://www.lmfdb.org/zeros/zeta/" target="_blank">https://www.lmfdb.org/zeros/zeta/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>