How to evaluate Riemann Zeta function

Assuming you're talking about numerical evaluation, along the critical strip $0<\Re s<1$ and "large" $\Im s$, (which is the region of interest for many) the Riemann-Siegel formula is standard; off the strip, what you can manage is a polyalgorithm.

For $\Re s\leq0$, one can use the reflection formula for $\zeta$,

$$\zeta(1-s)=\frac{2}{(2\pi)^s}\cos\left(\frac{s\pi}{2}\right)\Gamma(s)\zeta(s)$$

so we can consider evaluation for $\Re s>0$ in what follows.

Note that if $|s|$ is "large enough" (how large is "large" depends on the computing environment you're in), one can simply use the defining series $\zeta(s)=\sum\limits_{j=1}^\infty j^{-s}$, since its terms quickly diminish in magnitude. That leaves the problem of what to do for small to medium-sized $s$.

The key is to use the Dirichlet $\eta$ function:

$$\eta(s):=-\sum_{j=1}^\infty \frac{(-1)^j}{j^s}$$

which is related to $\zeta$ by the following identity:

$$\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$$

The reason for our interest in $\eta$ is that although this sum is slowly convergent, it is an alternating series, and a number of algorithms exist for quickly finding the sum of an alternating series numerically.

One method is the Levin transformation; another one, which is one of the simplest methods for numerically summing alternating series (and my personal favorite) is the Cohen-Rodriguez Villegas-Zagier algorithm. The algorithm is a bit too long to describe here, so I will just have to point you to the original paper.

This is in fact identical to the approach taken by Borwein in this paper.

You should look at the work of Wadim Zudilin. In particular you should look at "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational" (Turpion link, pdf 91k, gzip ps 80k) in Russian Math. Surveys 56:4 (2001), 774--776;