Algebraicity of L-values for GSP4 X GL2 and G

We prove algebraicity results for critical L-values attached to the group ${\rm GSp}_4 \times {\rm GL}_2$, and for Gan–Gross–Prasad periods which are conjecturally related to central L-values for ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$. Our result for ${\rm GSp}_4 \times {\rm GL}_2$ overlaps substantially with recent results of Morimoto, but our methods are very different; these results will be used in a sequel paper to construct a new p-adic L-function for ${\rm GSp}_4 \times {\rm GL}_2$. The results for Gan–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.