On the mean value of the generalized Dirichlet L-functions

Let q ⩾ 3 be an integer, let χ denote a Dirichlet character modulo q. For any real number a ⩾ 0 we define the generalized Dirichlet L-functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L(s,\chi ,a) = \sum\limits_{n = 1}^\infty {\frac{{\chi (n)}} {{(n + a)^s }},} $$\end{document} where s = σ + it with σ > 1 and t both real. They can be extended to all s by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet L-functions especially for s = 1 and s = 1/2 + it, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.