How to calculate non-trivial zeros of the Riemann zeta function

Question

I wanted to know how riemann calculated some non-trivial zero of the zeta function. Would I like a manual calculation.

Answer 1

Harold M. Edwards book 'Riemann's Zeta Function' covers Riemann's work on the zeta function in depth.

To answer your question, try:

Riemann Zeta Function, p.5

and Gram's method:

Riemann Zeta Function, p.96

Answer 2

One can calculate nontrivial zeros of $\zeta(s)$ along the line $s=1/2+t \in \mathbb{C}$ using the algorithm below.

$\mathbf{Newton-Raphson}$ $\mathbf{Algorithm}$ $\mathbf{for}$ $\zeta(s)$: Given an initial $t_{k} \in \mathbb{R}$, iterative solutions $t_{k+1}$ converge to nontrivial zeros of $\zeta(s)$,

$t_{k+1}=t_{k}-\frac{2 i}{\frac{16 i t_{k}}{1+4t_{k}^2}+\log_{e}(\pi) - \frac{\Gamma'(1/4+i t_{k}/2)}{\Gamma(1/4+i t_{k}/2)}-\frac{2 \zeta’(1/2+i t_k)}{\zeta(1/2+i t_k)}}$.

Note that $t_k$ is the imaginary part of the root $s=1/2+i t_k$, with $\zeta(1/2+it_k)\approx 0$.