Pullbacks of Saito--Kurokawa lifts and a central value formula

We prove an explicit central value formula for a family of complex $L$-series of degree $6$ for ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{L}}_3$ 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.