On special values of certain L-functions

We prove an algebraicity result concerning special values of certain automorphic $L$-functions. Namely let $f$ be a holomorphic newform of weight $k$ and let $\pi$ denote the attached irreducible unitary cuspidal representation of ${\rm GL}_2(\Bbb{A}_{\Bbb{Q}})$. Let $V$ be a quadratic space defined over $\Bbb{Q}$ such that $V\otimes_{\Bbb{Q}}\Bbb{R}$ is anisotropic. Let $\tau$ be an irreducible automorphic representation of ${\rm SO}(V,\Bbb{A}_{\Bbb{Q}})$. Then we prove an algebraicity result on the special value of $L_S(s,\pi \otimes\tau)$ at the rightmost critical point under some conditions. As a special case we prove a new algebraicity result on the special value of the Rankin triple $L$-function for ${\rm GL}(2)$ in some unbalanced case, which conforms with Deligne's conjecture on special values of motivic $L$-functions made explicit by Blasius in the aforementioned case.