On Periods and $L$-functions for $\mathbf{GL}_4 \times \mathbf{GL}_2$

We give a new integral representation of the $\wedge^2 \otimes \mathrm{std}_2$ $L$-function of generic cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$. In the former case, we use it to prove a relation between its central $L$-value and the generalized Shalika period. Exploiting the theta correspondence for $(\mathbf{GL}_4,\mathbf{GL}_4)$, we further establish a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $(\mathbf{GL}_4 \times \mathbf{GL}_2,\mathbf{GL}_2 \times \mathbf{GL}_2)$ and its corresponding period. In the case of cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ that are unramified everywhere, our formulas give new evidence towards conjectures of Wan-Zhang and of Gan-Gross-Prasad for $\mathbf{GSpin}_6 \times \mathbf{GSpin}_3$.