The Riemann Zeta function is defined as $$\zeta(s) = \sum_{n =1}^\infty \frac 1{n^s}$$
for $s \in \mathbb{C}$ and $\text{Re}(s) > 1$.
To solve for the real and imaginary parts of $\zeta(s)$ in this range, let $s = a + bi$:
$$\zeta(s) = \sum_{n =1}^\infty n^{-(a+bi)} = \sum_{n =1}^\infty e^{-(a+bi) \log(n)} = \sum_{n =1}^\infty e^{-a\log(n)}e^{i(-b\log(n))} \\ \implies \zeta(s) = \sum_{n =1}^\infty \frac{\cos{(b\log(n))}}{n^a} - i\sum_{n =1}^\infty \frac{\sin{(b\log(n))}}{n^a} \\ \\ \therefore \text{Re}(\zeta(s)) = \sum_{n =1}^\infty \frac{\cos{(b\log(n))}}{n^a}, \text{Im}(\zeta(s)) = -\sum_{n = 1}^\infty \frac{\sin{(b\log(n))}}{n^a} $$
With this, the complex output of $\zeta(s)$ can be visualized on a graphing calculator by translating $(x,y) \to (\text{Re}(\zeta(s)), \text{Im}(\zeta(s)))$. I did this myself using Desmos.
However, graphing this is quite taxing since these sums converge very slowly—about as slowly as the harmonic series. It takes about 10,000 terms for the series to converge to at least 4 decimal places of accuracy.
Are there any more efficient ways to graph the real and imaginary parts of $\zeta(s)$ in this manner? Can we speed up the convergence of the sums somehow, so that they converge to their respective value with fewer terms?