I have checked in [Andrew Odlyzko](http://www.dtc.umn.edu/~odlyZko/zeta_tables/index.html), Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I would like to know if there is an example of zeros as $s=0.5+at$ with $a$ a positive integer .



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> > **Now my question here is:**
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> > Could the complex zeros of the Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer?
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Thank you in advance.



Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer?
As explained by Peter Humphries in the comments, the famous conjecture here is that the imaginary part of the non-trivial zeros of $\zeta(s)$ is transcendental (or equivalently, linearly independent over $\mathbb{Q}$).


I'm not sure of the origins of the conjecture, but it was certainly around before the Rubinstein-Sarnak paper. Rumely a year before alredy says that it is "generally believed" for all Dirichlet L-functions. Both papers are avaible online:


* Robert Rumely, "[Numerical Computations Concerning the ERH](http://www.ams.org/mcom/1993-61-203/S0025-5718-1993-1195435-0/S0025-5718-1993-1195435-0.pdf)" (1993)
* Michael Rubinstein & Peter Sarnak, "[Chebyshev's bias](https://projecteuclid.org/euclid.em/1048515870)" (1994)


In particular, you can consider the fact that $k$ is not an integer as numerically verified at least in the range in which the RH has (up to $10^{13}$ after Gourdon).


