The Riemann Zeta Function is most commonly defined as $$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^s}$$ There is some sort of million dollar prize that involves proving the real part of complex number s must be $\frac{1}{2}$ for all nontrivial zeros. Of course this intregued me, because well, it's a million dollars. Odds are I won't solve it, but still. Anyway, I started looking at it and realized that you'd be raising a number to a complex power. This made no sense to me, so I went online and found Euler's formula that explains how that would work $$e^{i\pi}=-1$$ It turns out that the smallest nontrivial zeros is at about $\frac{1}{2}+14.1345i$, so I plugged it in to the zeta function. I used Desmos.com, and used separate summations for the real and imaginary parts. I expected to get zero. I did not get zero. In fact, the bigger I had the summation get, say, instead of summing to 1000000, I'd sum to 1000000000, the further off I would get from zero.
So tell me, how exactly are values for the Riemann Zeta Function computed?