Special values of L-functions

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

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 ${\displaystyle {\tfrac {1}{4}}}$ on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.