  The main purpose of this paper is to study higher order moments of the
generalized quadratic Gauss sums weighted by $L$-functions using estimates for
character sums and analytic methods. We find asymptotic formulas for three
character sums which arise naturally in the study of higher order moments of
the generalized quadratic Gauss sums. We then use these character sum estimates
to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of
the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic
formulas satisfy a conjecture of Wenpeng Zhang.
