Proving a known zero of the Riemann Zeta has real part exactly 1/2 Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$.  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 odd negative integers.  This doesn't affect the question but seemed worth correcting.  Thanks to the people who contributed.

You should prove that for example $ \frac{\xi(s)}{\xi(0)} = \frac{\det(H+1/4+s(s-1)}{\det(H+1/4)}$ with $H$ being a Hamiltonian operator with potential $ V^{-1}(x) $ proportional to the half-derivative of the Eigenvalue Staircase $ N(x)\pi = Arg \xi(1/2+ i \sqrt x) $  (see here).
Since the Hamiltonian operator is Hermitian then all the roots of the Riemann zeta have real part $ 1/2$. This proof is similar to the proof for the sine function
$ \frac{sin(x)}{x} $ with the potential $ V(x)=0 $ and boundary conditions $ y(0)=y(1)=0 $
