Here we see the modulus |ζ(z)| of the zeta function plotted over the complex plane where z = x + iy. Recall the definition |a + ib|² = |a|² + |b|². Note the pole at z = 1.

Unfortunately we cannot represent the behaviour of ζ(z) itself with a static graph, because it is a complex-valued function over the complex plane. It may be useful to think of it in more 'active' terms, acting on the complex plane to transport the point z to the point ζ(z).

However, a Mathematica application developed by Bernd Thaller (University of Graz) allows 2D or 3D representation of complex-valued functions through the use of colours. Here is a representation of the zeta function produced by Alex Astashyn using this application.