nt.number theory - citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function

Perhaps not an answer to your question, but certainly related.

In his Twelve Lectures, Hardy discusses the "Ramanujan hypothesis" to the effect that
$$
|tau(p)|le2p^{11over 2}
$$
for every prime $p$, and says that

this is the most fundamental of the unsolved problems
presented by the function.

He must have been talking about Ramanujan's 1916 paper in which he also conjectured the multiplicativity and the congruences for the $tau$-function.

The multiplicativity was established by Mordell (1918) using what we would today call Hecke operators, the congruences were studied by Swinnerton-Dyer and Serre in the early 70s, ultimately leading Serre to his modularity conjecture as proved recently by Khare--Wintenberger (2009), and the estimate $|tau(p)|le2p^{11over 2}$ followed from Deligne's proof (1973) of the Weil conjectures.

Not bad as far as the mathematics inspired by a single paper goes.

Addendum. The estimate $|tau(p)|le2p^{11over 2}$ appears as formula (104) on page 153 of Ramanujan's Collected Papers as being ``highly probable''.