Let $s$ be a complex number.
Define $\zeta(s)$ as the analytic continuation of
$$\zeta(s) = \frac{\prod_p \dfrac{1}{1 - p^{-s}} + \prod_q \dfrac{1}{1 - q^{-s}}}{2} $$
where $p$ are the primes $1 \mod 4$ and $q$ are the primes $3 \mod 4$.
Where are the non-real zero's of this function ?
$$\zeta(s) = 0 $$
What is the smallest and largest real part of the non-real zero's ?