A query regarding the non-trivial zeroes of Zeta function?

The zeta function in its functional form is described by:

$$\zeta(s) = 2^s\pi ^{s-1}\sin\left( \frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$$

We know that the zeroes of the zeta function are symmetric about the real line $$1/2$$.

Can there be two zeroes (whose real part is asymmetric about the 1/2 axis) such that: $$|(\Re(s_0)|\neq|\Re(s_1)|$$ but $$\Im(s_0)=\Im(s_1)$$.

I could find no information regarding the same?

• I think it is unknown. Indeed what would it change if it was true or not, assuming the RH fails ? If you can hardly see any noticeable consequence of it, you can hardly prove or disprove it is true. To me the main question assuming the RH fails is if $\sup_\rho \Re(\rho)$ is attained, but finitely or infinitely many zeros, or if it is not attained, in that case, what $n(T,\epsilon) =\# \{ \rho, |\Im(\rho)| < T, \Re(\rho) > \epsilon\}$ looks like and does $n(T,\frac{A}{\log T}) \to 0$ – reuns Feb 9 at 19:52
• I am not sure the question is correct as obviously if RH fails, there are zeros with different real parts in absolute value (by the 1/2 symmetry, zeroes off the critical line come in 4's) and same imaginary part. Maybe you wanted this question asked with different real parts but both greater than a half? Or same real part (not a half) but different positive imaginary parts? As noted above, nothing is known about non-trivial zeros off the symmetry line even if they exist - the critical line density of zeroes (100% expected anyway)/Lindelof are probably the most interesting questions if RH fails – Conrad Feb 9 at 20:22
• "I am not sure the question is correct as obviously if RH fails, there are zeros with different real parts in absolute value (by the 1/2 symmetry, zeroes off the critical line come in 4's) and same imaginary part. " - true. I was essentially asking about the zeroes whose real part is asymmetric about the 1/2 axis (the symmetric real ones are obvious if RH is false). Updating. – TheoryQuest1 Feb 9 at 20:40