  Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of
level one and weight $ 4k $, $ k \in N $. We prove an explicit formula for the
mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the
symmetric square $L$-function $ L(sym^2\mathfrak{f}, 1/2)$, relating it to the
dual mixed moment of the double Dirichlet series and the Riemann zeta function
weighted by the ${}_3F_{2}$ hypergeometric function. Analysing the
corresponding special functions by the means of the Liouville-Green
approximation followed by the saddle point method, we prove that the initial
mixed moment is bounded by $\log^3k$.
