For example if you look at this graph the real and imaginary parts along the critical line $x$ is plugged into what equation? It can't be the normal function:
$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$
my understanding is that that is divergent in the critical strip. I know enough to know that you have to analytically continue the normal function, but reading all this different stuff there's all kinds a weird series and confusion. Perhaps I don't know enough to even ask my question, but for them to have plotted that graph up there don't they have some functions to plug x into? So for example was it this one? 
or is it this? 
or perhaps it was this guy: 
Do you see what I'm saying? Where's the y=f(x) versions that make those plots of zeta going to zero along re=1/2? I feel like knowing how to plot that would help me understand the Riemann hypothesis a little better.
Best Answer
As a random example, the Boost C++ libaries (v. 1.51.0) have a
zeta()implementation. Suppose you wish to compute $\zeta(s)$ to within some (small) target precision. They start by, if necessary, using the reflection formula to ensure $\mathrm{Re}(s) \geq 0$. Then they choose the parameter $n$ so that $\frac{1}{8^n |1 - 2^{1-s}|}$ is smaller than your target precision. They use $$ \zeta(s) = \frac{-1}{s^n(1-2^{1-s})} \sum_{j = 0}^{2n-1} \frac{e_j}{(j+1)^s} + \gamma(s) \text{,} $$ where $$ e_j = (-1)^j \left(\sum_{k=0}^{j-n} \frac{n!}{k!(n-k)!} - 2^n \right) \text{.} $$ Then they compute and return $$ \frac{-1}{s^n(1-2^{1-s})} \sum_{j = 0}^{2n-1} \frac{e_j}{(j+1)^s} \text{.} $$Since it is known that $|\gamma(s)| \leq \frac{1}{8^n |1 - 2^{1-s}|}$, the difference between the computed term and the exact value of $\zeta(s)$ is smaller than your target precision.