On the new identities of Dirichlet $L$-functions

TL;DR

This paper derives new identities involving Dirichlet $L$-functions and Gauss sums, providing specific formulas for $L(2, ext{chi})$, enhancing understanding of their special function series.

Contribution

It introduces novel identities for Dirichlet $L$-functions involving Gauss sums, including explicit formulas for $L(2, ext{chi})$, expanding the analytical tools available.

Findings

New identities for Dirichlet $L$-functions involving Gauss sums

Explicit formulas for $L(2, ext{chi})$

Enhanced analytical understanding of special function series

Abstract

Let $q\ge3$ be an integer, $\chi$ be a Dirichlet character modulo $q$, and $L(s,\chi)$ denote the Dirichlet $L$-functions corresponding to $\chi$. In this paper, we show some special function series, and give some new identities for the Dirichlet $L$-functions involving Gauss sums. Specially, we give specific identities for $L(2,\chi)$.