Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions

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 $\log^3k$.