The Second Moment of ${\mathrm{G}\mathrm{L}}_3\times {\mathrm{G}\mathrm{L}}_2$ $L$-functions at Special Points

Let $\phi$ be a fixed Hecke–Maass form for ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ and $u_j$ traverse an orthonormal basis of Hecke–Maass forms for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. Let $1\mathrm{/}4+t_j^2$ be the Laplace eigenvalue of $u_j$. In this paper, we prove the mean Lindelöf hypothesis for the second moment of $L(1\mathrm{/}2+i{t}_j,\phi \times u_j)$ on $T<t_j⩽T+\sqrt{T}$. Previously, this was proven by Young on $t_j⩽T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein–Kloosterman’ cancellation.