In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:

If $\phi_n$'s are orthonormal Hecke-Maass eigenforms in some arithmetic congruence (compact/noncompact) manifold $M=\Gamma\backslash G$, i.e. normalized eigenforms of Laplacian and Hecke operators in corresponding manifold, then for $I\subset\mathbb{N}$, $$\left(\int_M\prod_{n\in I}\phi_ndvol_M\right)^2=C\frac{L\left(\frac12,\bigotimes\limits_{n\in I}\phi_n\right)}{\prod\limits_{n\in I}L(1,sym^2\phi_n)},$$ for some constant $C$.

$\textbf{Question}:$

1) Is the above (or some modified version) true?

2) If not, why is this only true for THREE eigenforms?

Thanks. Any help or reference is highly appreciated.