On the new identities of Dirichlet $L$-functions

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 $L(2,\chi)$.