Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a 0 + a 1 x + ... + a k x k is a k-degree polynomial with integral coefficients such that (p, a 0, a 1, ..., a k ) = 1, for any integer m, we study the asymptotic property of $$ \sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}} {p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } , $$ where e(y) = e2πiy . The main purpose is to use the analytic method to study the 2m-th power mean of Dirichlet L-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.