On Petersson norms of generic cusp forms and special values of adjoint $L$-functions for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint $L$-functions and some elementary constants depending only on local representations.