  Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$,
for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions
$$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where
$s=\sigma+it$ with $\sigma>1$ and $t$ both real. It can be extended to all $s$
by analytic continuation. For any integer $m$, the famous Gauss sum $G(m,\chi)$
is defined as follows:
$$G(m,\chi)=\sum_{a=1}^{q}\chi(a)e\left(\frac{am}{q}\right), $$ where
$e(y)=e^{2\pi iy}$. The main purpose of this paper is to use the analytic
method to study the mean value properties of the generalized Dirichlet
$L$-functions with the weight of the Gauss Sums, and obtain a sharp asymptotic
formula.
