  We prove an explicit central value formula for a family of complex $L$-series
of degree $6$ for $\mathrm{GL}_2 \times \mathrm{GL}_3$ which arise as factors
of certain Garret--Rankin triple product $L$-series associated with modular
forms. Our result generalizes a previous formula of Ichino involving
Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture
stating the algebraicity of the central values of the considered $L$-series up
to the relevant periods.
