Riemann Zeta function, nontrivial zeroes

Question

How can we prove what are, say the first 4 non-trivial zeroes of the Riemann $\zeta$ on the critical line $Re(z_j)=\frac{1}{2}$, $j=1,2,3,4$ the first two with negative imaginary part and the second two with positive imaginary part? What can in general be said about the frequency of the imaginary part with which all the non-trivial zeroes occur, if something at all?

Answer 1

As for computing the zeros, there's a good discussion in "Riemann's Zeta Function," by H. M. Edwards. I don't understand what you mean by "the frequency of the imaginary part." No imaginary part can occur more than once (unless the hypothesis is false.)

Here is a student paper on the subject that probably owes a lot to Edwards's discussion.