In this paper, we consider the zeta function $Z(P,\chi,s)$ associated with a polynomial $P(X)\in {\Bbb R}[X_1,\ldots,X_r]$ and $\chi=(\chi_1,\ldots,\chi_r)$ with $\chi_j$ non-trivial Dirichlet characters, defined by $$ Z(P,\chi,s) = \sum^\infty_{n_1=1}\cdots\sum^\infty_{n_r=1}\chi_1(n_1)\cdots\chi_r(n_r) P(n_1,\ldots,n_r)^{-s}, $$ which is absolutely convergent for sufficiently large Re\,$s$ under some conditions on $P(X)$. We shall prove that the special value $Z(P,\chi,-m)$ is completely determined by $P^m(X)$ in a simple way. As an immediate application, we give a closed expression for sums of products of any number of generalized Bernoulli numbers.