newton’s method to find complex fixed points of zeta(s)=s

The red corresponds to a real fixed point of the Riemann zeta function $\zeta(s)$ where $s\approx 1.83377265168027139624564858944\ldots$, and the cyan corresponds to another real fixed point, with $s\approx -0.295905005575213955647237831083\ldots$

It’s worth mentioning that there’s a paper Riemann’s zeta function and Newton’s method by Tomoki Kawahira that talks about the application of Newton’s method to root finding on the zeta function, but does not address the use of Newton’s method to visualize the process of finding fixed points of the Riemann zeta function.