How are Zeta function values calculated from within the Critical Strip?

An extension of the area of convergence can be obtained by rearranging the original series. The series

$$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)$$

converges for $\Re s > 0$. See here.

This is an outline of the steps taken in the ref. in the comments to derive an expression for $\zeta(s)$ which converges for $\sigma>0$ and which can be modified slightly to allow approximation of values of $\zeta(s).$

Letting $f(s) = 1/x^s$ and $\int_0^{\infty}\frac{dx}{x^s}=\frac{1}{s-1}$ for $\text{Re}(s)>1$ and using (proof omitted)

$$\sum_1^{\infty}f(r)-\int_1^{\infty}f(t)dt = f(1)+\int_1^{\infty}(t-[t])f'(t)dt$$

we get

$$(*)\hspace{10mm}\zeta(s)=\frac{1}{(s-1)}+1-s\int_1^{\infty}\frac{x-[x]}{x^{s+1}}dx. $$

Since $x - [x] < 1$ the integral on the right converges for all $\sigma > 0.$

The above is of no use for computation so using Euler's summation formula for partial sums,

$$ \sum_{n=2}^{\infty}f(n) = \int_1^N f(x)dx+\int_1^N(x-[x])f'(x)dx,$$

applying this to $f(x)= 1/x^s$ in which $(s \neq 1)$ we have

$$\sum_{n=1}^N\frac{1}{n^s}=1 + \frac{1}{s-1}-\frac{N^{1-s}}{s-1}-s\int_1^N\frac{x-[x]}{x^{s+1}}dx $$

in which the term $1$ is just $f(1).$

Subtracting this from (*) we get

$$\zeta(s) = \sum_{n=1}^N \frac{1}{n^s}+\frac{N^{1-s}}{s-1} +r(s)$$

in which the term $r(s)$ can be shown to vanish as $N\to\infty.$

And so we have

$$\zeta(s) = \lim_{N\to\infty}\left(\sum_{n=1}^N\frac{1}{n^s}+\frac{N^{1-s}}{s-1} \right)$$

Now if I use this to calculate $\zeta(1+it)$ along the imaginary axis from $t = 0$ to $t = 36$ as Jameson does in his text, or other values, I get good approximations for even low values of N. For $s= 3/4+i $ and $N=1000$ I get $\zeta(3/4+i)\approx 0.33-.87i$ which agrees with $Mathematica's$ figure pretty closely.

The list of results omitted to do this is not long and they are all furnished in the noted reference.

Although unmentioned by the others, once you have a functional equation, you can also use a so-called "approximate functional equation". In this case, it may seem circular, because knowing the functional equation implicitly involves understanding the values in the center of the strip.

But in fact, the approximate functional equation is an extremely efficient method of approximating values in the critical strip. A reference on wikipedia the Riemann-Siegel formula. A slightly looser explanation is that for $s = \sigma + it$, we have $$\zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \gamma(1-s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}),$$ where $\gamma(s)$ is a ratio of Gamma functions and you can choose $x,y$ to give the approximations of desired size. This works poorly on the edges of the critical strip, but very well in regions bounded away from the edges. It works especially well on the critical line, which is of special interest.