Since we have
$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$
where $\vartheta(t)$ is the Riemann Siegel function.
we may deduce (with some effort) that there is no other zero, and those on the critical line have ordinates the zeros of
the cosine or sine of the real function
$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$