The question is in the title: if I'm not mistaken, what misses to prove that all automorphic L-functions belong to the Selberg class is a proof of the Ramanujan Conjecture. But the Selberg class is expected to be the natural class of L-functions that satisfy the analogue of the Riemann Hypothesis.

Moreover, Ramanujan Conjecture for automorphic L-functions states that, if $\pi$ is an irreducible unitary cuspidal representation of $GL_{m}(\mathbb{Q}_{\mathbb{A}})$, then for any unramified prime $p$, $\vert\alpha(p,j)\vert=1$. The striking fact is that this conjecture is about complex numbers related to a given automorphic L-function that are supposed to lie on the unit circle, whereas the Grand Riemann Hypothesis is about other complex numbers related to this automorphic L-function that are supposed to lie on a straight line.

But we know that, through homographies, that are rather natural transformations of the Riemann sphere, straights lines and circles are essentially the same objects. Furthermore, I guess that, through Hadamard's factorization theorem, one can expect to have a rather straightforward expression of the non trivial zeroes of a given automorphic L-function in terms of all its $\alpha(p,j)$.

So, would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis?

Thanks in advance.

expectedto satisfy RH and that RC would put the L-functions in question into the Selberg class. But why should this imply theyactuallysatisfy RH. It is only expected after all, and not known for L-functions very well-known to be in the Selberg class either. $\endgroup$ – user9072 May 28 '13 at 17:01