Algebraicity of $L$-values for ${\text{GSp}}_4\times {\text{GL}}_2$ and ${\text{GSp}}_4\times {\text{GL}}_2\times {\text{GL}}_2$

We prove algebraicity results for critical $L$-values attached to the group ${\text{GSp}}_4\times {\text{GL}}_2$, and for Gan–Gross–Prasad periods which are conjecturally related to central $L$-values for ${\text{GSp}}_4\times {\text{GL}}_2\times {\text{GL}}_2$. Our result for ${\text{GSp}}_4\times {\text{GL}}_2$ gives a new proof (by a very different method) of a recent result of Morimoto, and will be used in a sequel paper to construct a new $p$-adic $L$-function for ${\text{GSp}}_4\times {\text{GL}}_2$. The results for Gross–Prasad periods appear to be new. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note.