Special values of L-functions

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the study of special values of L-functions is a subfield of <a href="/facts/Number_theory/zhoa7zrc">number theory</a> devoted to generalising formulae such as the <a href="/facts/Leibniz_formula_for_pi/xkF1WgLD">Leibniz formula for π</a>, namely

1
        
        −
        
        
          
            1
            3
          
        
        
        +
        
        
          
            1
            5
          
        
        
        −
        
        
          
            1
            7
          
        
        
        +
        
        
          
            1
            9
          
        
        
        −
        
        ⋯
        
        =
        
        
          
            π
            4
          
        
        ,
        
      
    
    {\displaystyle 1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \;=\;{\frac {\pi }{4}},\!}

</p><p>by the recognition that expression on the left-hand side is also 
  
    
      
        L
        (
        1
        )
      
    
    {\displaystyle L(1)}
  
 where 
  
    
      
        L
        (
        s
        )
      
    
    {\displaystyle L(s)}
  
 is the <a href="/facts/Dirichlet_L-function/my8hc6Qy">Dirichlet L-function</a> for the field of <a href="/facts/Gaussian_rational/X0jN1MnI">Gaussian rational</a> numbers. This formula is a special case of the <a href="/facts/Analytic_class_number_formula/0GBPnrwS">analytic class number formula</a>, and in those terms reads that the Gaussian field has <a href="/facts/Class_number_problem/bIrLDjde">class number 1</a>. The factor 
  
    
      
        
          
            
              1
              4
            
          
        
      
    
    {\displaystyle {\tfrac {1}{4}}}
  
 on the right hand side of the formula corresponds to the fact that this field contains four <a href="/facts/Roots_of_unity/EqH0ho1Y">roots of unity</a>.
</p>

Special values of L-functions open-in-new

Special values of L-functions