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The main purpose of this paper is to study the hybrid mean value of $\frac{L^{\text{'}}}{L}(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value ${\sum }_{\chi \ne {\chi }_0}\left|\tau \left(\chi \right)\right||\frac{L^{\text{'}}}{L}{(1,\chi )|}^{2k}$ of $\frac{L^{\text{'}}}{L}$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.

Let $q\ge 3$ be an integer, let $\chi$ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac{\chi \left(n\right)}{{(n+a)}^s},$$ where $s=\sigma +\mathrm{i}t$ with $\sigma >1$ 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 $s=\frac{1}{2}+\mathrm{i}t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.