We consider a cross-ratio of Rankin--Selberg $L$-functions associated to automorphic representations $\Sigma,\Sigma'$ of ${\rm GL}_n$ and $\Pi,\Pi'$ of ${\rm GL}_n'$:
\[
\frac{L(s,\Sigma \times \Pi)\, L(s,\Sigma' \times \Pi')}{L(s,\Sigma \times \Pi')\, L(s,\Sigma' \times \Pi)}.
\]
When the automorphic representations are algebraic and satisfy $\Sigma_\infty = \Sigma'_\infty$ and $\Pi_\infty = \Pi'_\infty$, we conjecture that this ratio is algebraic and Galois-equivariant at critical points, reflecting the cancellation of transcendental periods.
In this talk, we introduce applications of the cross-ratio formula to the algebraicity of critical $L$-values, including: the symmetric power and tensor product $L$-functions for ${\rm GL}_2$; the tensor product $L$-functions for ${\rm GL}_n \times {\rm GL}_2$ and ${\rm GSp}_4 \times {\rm GSp}_4$; and Blasius' conjecture on Artin twists.