$\DeclareMathOperator\GL{GL}$Typical $\GL(3)$ automorphic L-functions are often depicted as looking like the following Dirichlet series:
$$L(s, \pi) = \sum_{n >0} \frac{A(1,n)}{n^s}$$

Is there a specific reason for the L-function to be of this form (rather than with $A(n,1)$ say, or $A(n,n)$ or $A(n,m)$?). I would like to see a link with the classical cases of the zeta function or of $\GL(2)$, where it comes from the Fourier expansion and defining the L-function as a Mellin transform of the form itself.