Summary: In this paper, we announce the result of the joint work [Am. J. Math. 138, No. 4, 1117–1166 (2016; Zbl 1404.11064)] with M. Furusawa, in which we state a conjecture (see Conjecture 2) on the algebraicity of special values of tensor product \(L\)-functions for automorphic representations of \(\operatorname{SO} (V)\) and \(\operatorname{GL}_2\) explicating Deligne periods and we show this conjecture at various critical points under certain assumption on the weight (see Theorem 3.1). This algebraicity result is a generalization of our previous result [M. Furusawa and the author, ibid. 136, No. 5, 1385–1407 (2014; Zbl 1317.11050), Theorem 1] with respect to critical points and infinity types of automorphic representations of \(\operatorname{SO} (V)\).