The Euler numbers and recursive properties of Dirichlet L-functions

The aim of this paper is using an elementary method and the properties of the Bernoulli polynomials to establish a close relationship between the Euler numbers of the second kind $E_n^{\ast }$ and the Dirichlet L-function $L(s,\chi )$. At the same time, we also prove a new congruence for the Euler numbers $E_n$. That is, for any prime $p\equiv 1mod8$, we have $E_{\frac{p-3}{2}}\equiv 0modp$. As an application of our result, we give a new recursive formula for one kind of Dirichlet L-functions.