Abstract: By using estimation of the classical kloostermann sum,estimation of the trigonomitric sum and the analytic method to study the first power mean of Dirichlet L-functions with the weight of general Kloostermann sum,a sharp asymptotic formula is given:∑x≠x0｜S(m,n,x,q)｜2｜L(1,x)｜=φ2(q)∑′∞n=1(r2(n))/(n2)+O(q3/2+ε).

Key words: Dirichlet L-functions, distribution of the mean value, general Kloostermann sum, asymptotic formula