Let frakturf run over the set H4k of primitive cusp forms of level one and weight 4k, k∈N. We prove an explicit formula for the mixed moment of the Hecke L‐function L(f,1/2) and the symmetric square L‐function L(prefixsym2f,1/2), relating it to the dual mixed moment of a double Dirichlet series and the Riemann zeta function weighted by the 3F2 hypergeometric function. Analysing the corresponding special functions by means of the Liouville–Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded above by log3k.