  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.
