Like the moment problem, nothing is known
about infinitude of zeroes on the critical line for
degree d > 2 L-functions.
In fact, aside from zeroes at s = 1/2 coming
from algebraic geometry, it is not known there
are any zeroes on the critical line for
d > 2.
Possible approach: if a certain Fourier
component of the automorphic distribution of a
) vanishes to
infinite order at
, then L(1/2+it,
) = 0 for
infinitely many t