The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ of level $N>1$ and eigenvalue $1/4+\lambda^2$, will Lindelof Hypothesis imply Selberg Eigenvalue conjecture, i.e., $\lambda\geq 0$?
I know that functoriality results such as Gelbart-Jacquet, Shahidi, Kim gives a very good bound, such as Kim-Sarnak bound of 7/64. Obviously functoriality for of all symmetric powers implies the Selberg eigenvalue conjecture. The proof involves some non-vanishing results for families of $L$-functions (Kim-Sarnak, Luo-Rudnick-Sarnak). I am thinking about the relation of the Selberg eigenvalue conjectures and the GRH/Lindelof.
Lastly, GRH itself at a single point for one L-function probably does not say much about the Selberg eigenvalue conjecture because GRH only excludes non-trivial zeros. A violation of Selberg eigenvalue conjecture will give us a zero from Gamma factors (thus it is trivial zero). But family of L-functions with GRH or Lindelof possibly can tells more about non-vanishing (trivial zeros) of L-functions.