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 $\log^3k$.
Categories: math.NT math.CA