  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 $L(2,\chi)$.
