Simultaneous non-vanishing of central values of $\mathrm{G}\mathrm{L}(2)\times \mathrm{G}\mathrm{L}(3)$ and $\mathrm{G}\mathrm{L}(3)\times \mathrm{G}\mathrm{L}(3)$ $L$-functions

Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$ on ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, and let $\pi$ be a fixed cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_3$. In this paper, we establish an asymptotic formula for the first moment of the product $$L(1\mathrm{/}2,g\times F)L(1\mathrm{/}2,\pi \times F),$$ where $F$ runs over an orthonormal basis of Hecke-Maaß cusp forms of level $q$ on ${\mathrm{G}\mathrm{L}}_3$. As an application, we deduce that $L(1\mathrm{/}2,g\times F)L(1\mathrm{/}2,\pi \times F)\mathrm{\ne }0$ for infinitely many such forms $F$.