  We prove Deligne's conjecture for central critical values of certain
automorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$. The proof is
base on rationality results for central critical values of triple product
$L$-functions, which follow from establishing explicit Ichino's formulae for
trilinear period integrals for Hilbert cusp forms on totally real etale cubic
algebras over $\mathbb{Q}$.
