Visualizing the Riemann zeta function and analytic continuation

While working on (well, procrastinating from) an "Essence of Calculus" video, I was playing around with visualizing various complex functions.  When I put in the zeta function, it struck me as so pretty that I just had to make it the next project.  This was especially true when I realized it could give a nice way to motivate the idea of analytic continuation, which is one of the more opaque aspects of this function that stalls people trying to read more into it.

Hopefully, this can help motivate people to learn about complex analysis more generally.