Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.

Shimura have the following formula $L(s, Ad\; \phi)=\zeta(2s)\sum\frac{a(n^2)}{n^s}$. [*]

I would like to see its generalization to $GL(3)$ or $GL(N)$. Let $\phi$ be a $GL(3)$ automorphic form with Fourier coefficients $a(m,n)$ and $a(1,1)=1$. We have $L(s,\phi)=\sum \frac{a(1,n)}{n^s}$. I would like to know a formula for $L(s, Ad\; \phi)$ like [*].

We shall note that we always have $L(s,Ad\;\phi)=L(s,\phi \times \overline{\phi})/\zeta(s)$ but that is not what I want.