Let $F(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the Gamma-complete version of the Riemann $\zeta$ function.
Let $f(z)=F(1/2+i z)$. So it is known that all zeros of $f(z)$ are in the strip $S_{1/2}$ where $S_\delta=\{z\in\mathbb{C},|\operatorname{Im}(z)|\lt \delta\}$.
The Riemann Hypothesis is equivalent to that all zeros of $f(z)$ are real.
(A) Suppose that there exists a sequence of functions $\{f_n(z)\}_{n=0}^{\infty}$ such that (i) they are analytic in the strip $S_{1}$ and (ii) all but finite zeros of $f_n(z)$ are real.
(B) Suppose that this sequence of function $\{f_n(z)\}_{n=0}^{\infty}$ uniformly converge to $f(z)$ in the strip $S_{1}$.
Question: Is (A) and (B) enough for us to deduce, using Hurwitz's theorem, that all but finite zeros of $f(z)$ are real?
Any references and comments are welcomed.
Thanks- mike