In the present paper, the assumptions on the function $F(s)$ are more restrictive but the conclusions about the zeros are stronger in two respects: the lower bound for $\sigma$ can be taken closer to $\frac{1}{2}-C(\log\log T)(\log T)^{-1}$ and the lower bound for the number of zeros is like $T/\log\log\log T$.
@article{hal-01104799,
author = {Balasubramanian, R and Ramachandra, K},
title = {On the zeros of a class of generalised Dirichlet series-IX},
journal = {HAL},
volume = {1991},
number = {0},
year = {1991},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01104799}
}
Balasubramanian, R; Ramachandra, K. On the zeros of a class of generalised Dirichlet series-IX. HAL, Tome 1991 (1991) no. 0, . http://gdmltest.u-ga.fr/item/hal-01104799/