If non-trivial zeroes of the Riemann Zeta function are irrational, and we have to calculate $\zeta[t]$ numerically, how is it ever possible to prove that a particular zero of $\zeta[t]$ falls exactly on the $Re[t]=1/2$  line?  Are we really only able to prove that they're within our computational precision of that line?
Much effort has been expended on a famous unsolved problem about the .  Not surprisingly, it's called the Riemann hypothesis, which asserts:  $$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or } \operatorname{Im} s = 0 .$$  Now there are numerical methods for approximating $\zeta(s)$, but as I understand, no one knows any exact values except at even integers, which include the trivial zeroes for which $\operatorname{Im} s = 0 $.  So I've always wondered:  For the rest, how does one prove $\operatorname{Re} s = 1/2$ exactly?  (All I know that seems vaguely useful is the argument principle, which I'm not sure helps here, but I'd be happy to learn about other techniques that aren't too far advanced.)  Edit:  Looking over this again, I found out I missed the closed-form values at .  This doesn't affect the question but seemed worth correcting.  Thanks to the people who contributed.
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