Algebraicity of L-values for GSp4 × GL2 and GSp4 × GL2 × GL2

We prove algebraicity results for critical L-values attached to the group GSp₄ × GL₂, and for Gan–Gross–Prasad periods which are conjecturally related to central L-values for GSp₄ × GL₂ × GL₂. Our result for GSp₄ × GL₂ 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 GSp₄ × GL₂. 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.