sequences and series - How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist:

Given the zeros of the Riemann zeta function,

$\zeta(s)
= \sum_{n=1}^\infty n^{-s}$,
how do I actually evaluate them?

On this web page I found a list of zeros. Well I guess if the values, call one ${a_\text{zero}}$, are just given in decimal expansion, then I'd have to run a program and see how it approaches zero
$$\zeta({a_\text{zero}}) =
\sum_{n=1}^\infty n^{-{a_\text{zero}}} =
\frac{1}{1^{a_\text{zero}}} + \frac{1}{2^{a_\text{zero}}} + \frac{1}{3^{a_\text{zero}}} + \cdots\approx 0. \qquad ;\Re(a)>1$$

But are there also analytical solutions, let's call one ${b_\text{zero}}$, which I can plug in and see $\zeta({b_\text{zero}}) = 0$ exactly? Specifically the non-trivial ones with imaginary part are of interest. What I find curious is that these series, with each term being a real number multiplied by some $n^{-i\cdot \text{Im}(b_\text{zero})}$, then also kind of resemble Fourer series.