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Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums