: Mathématiques: special values of L-functions

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mathematical analysis > complex analysis > meromorphic function > L-function > special values of L-functions

special values of L-functions

In mathematics, the study of special values of $L$ -functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely ${\\displaystyle 1\\,-\\,{\ rac {1}{3}}\\,+\\,{\ rac {1}{5}}\\,-\\,{\ rac {1}{7}}\\,+\\,{\ rac {1}{9}}\\,-\\,\\cdots \\;=\\;{\ rac {\\pi }{4}},\\!}$ by the recognition that expression on the left-hand side is also ${\\displaystyle L(1)}$ where ${\\displaystyle L(s)}$ is the Dirichlet $L$ -function for the field of Gaussian rational numbers. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1. The factor ${\\displaystyle {\ frac {1}{4}}}$ on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Special_values_of_L-functions)