This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?
The functional equation of the zeta function gives you the trivial zeros at once:
if you input now $\,s=-2k\;,\;\;k\in\Bbb N\,$ , you get
Question: Why in the negative even integers and not in the positive ones? Hint: check stuff about the gamma function.
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Riemann did it - that's how he came up with his famous hypotheses. Of course, it was very arduous and he had to derive some pioneering results to do it, but that's why he is famous and I am not.
Do a search for "computing the Riemann zeta function" and you will get many hits. A good reference is Edwards' book "Riemann's Zeta Function".