Particular values of the Riemann zeta function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> is a function in <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, which is also important in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>. It is often denoted 
 
 
 
 ζ
 (
 s
 )
 
 
 {\displaystyle \zeta (s)}
 
 and is named after the mathematician <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a>. When the argument 
 
 
 
 s
 
 
 {\displaystyle s}
 
 is a <a href="/facts/Real_number/R02gw5Pb">real number</a> greater than one, the zeta function satisfies the equation

ζ
        (
        s
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            1
            
              n
              
                s
              
            
          
        
        
        .
      
    
    {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.}

It can therefore provide the sum of various convergent <a href="/facts/Series_(mathematics)/yDnlsgIe">infinite series</a>, such as 
 
 
 
 ζ
 (
 2
 )
 =
 
 
 1
 
 1
 
 2
 
 
 
 
 +
 
 
 {\textstyle \zeta (2)={\frac {1}{1^{2}}}+}

1
            
              2
              
                2
              
            
          
        
        +
      
    
    {\textstyle {\frac {1}{2^{2}}}+}

1
 
 3
 
 2
 
 
 
 
 +
 …
 
 .
 
 
 {\textstyle {\frac {1}{3^{2}}}+\ldots \,.}
 
 Explicit or numerically efficient formulae exist for 
 
 
 
 ζ
 (
 s
 )
 
 
 {\displaystyle \zeta (s)}
 
 at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in 
 
 
 
 s
 
 
 {\displaystyle s}
 
 above also holds when 
 
 
 
 s
 
 
 {\displaystyle s}
 
 is a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> whose <a href="/facts/Real_part/2w7mqI7e">real part</a> is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> by <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a>, except for a <a href="/facts/Simple_pole/gnA3U8VP">simple pole</a> at 
 
 
 
 s
 =
 1
 
 
 {\displaystyle s=1}
 
. The <a href="/facts/Complex_derivative/kVzthtEf">complex derivative</a> exists in this more general region, making the zeta function a <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic function</a>. The above equation no longer applies for these extended values of 
 
 
 
 s
 
 
 {\displaystyle s}
 
, for which the corresponding summation would diverge. For example, the full zeta function exists at 
 
 
 
 s
 =
 −
 1
 
 
 {\displaystyle s=-1}
 
 (and is therefore finite there), but the corresponding series would be 
 
 
 
 1
 +
 2
 +
 3
 +
 …
 
 ,
 
 
 {\textstyle 1+2+3+\ldots \,,}
 
 whose <a href="/facts/Partial_sum/yDnlsgIe">partial sums</a> would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>.

Particular values of the Riemann zeta function open-in-new

Particular values of the Riemann zeta function