Zeta, its Mandelbrot set, the root Julia set of c=0 and the Julia set of Newton's method, converging to the zeta zeros.
I have just develped a natty research application for Mac using XCode, which is a state of the art viewer and analytical tool for investigating the notorious Riemann zeta function and its Mandelbrot and Julia sets. You can download and explore this viewer and its source code and related research at:
Thanks to Mike Lewinski (see comments) who noticed it was not working in Snow Leopard this application and the other three associated fractal applications, including the twin application DHViewer which looks at the dynamics of rational and transcendental complex functions now should all be working in all systems from Tiger to snow Leopard.
RZViewer contains all the bells and whistles in terms of interactive menus check boxes and input and output necessary to manipulate and investigate the Riemann zeta function and its fractal properties and those of a spectrum of related Dirichlet L-functions which share the same enigmatic zeros.
These are very strange functions, unlike the trigonometric, exponential, logarithmic, rational and polynomial functions we are used to dealing with in calculus. They require massive computation and are complicated to program as they are represented as infinite series and products of complex numbers, so it was somewhat of a victory to find that the coding could be done directly using the gcc C library complex.h.
The viewer exposes in ultra-fast code many features of these functions which have not to my knowledge been explored before. Many of the images are intriguingly different from the fractal sets of standard functions we are more used to, although they do share some universal fractal properties.