The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$
and extends to the whole of $\mathbb{C}$ by analytic continuation. It is known that all the complex zeros of $\zeta(s)$ satisfy $0<\Re(s) <1$.
But am not sure how on they are calculated ?
Basically, naming $s=\sigma+it$, the idea is to use the function $$ \xi(s)=\Gamma(s/2)\pi^{-s/2}(s-1)\zeta(s) $$ because it's real-valued on the critical line $t=1/2$, hence you'll find a zero whenever $\xi(1/2+it)$ changes sign. There are various method to do that, a very nice introduction can be found in Edwards' book ``Riemann's Zeta Function", see for example Section 6.5.