Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e))

[Gottfried]

Hmm, I'm already losing the structure of the current threads, which focus the same things from different views. May be it's a good place here.

I was searching for a method, to find t where t^(1/t) = s, given s, where s is an arbitrary real number outside the range e^(-e)..e^(1/e). The h()-function derived from the python-implementation for Lambert-W in wikipedia seems to be valid only for the "inside bounds" case.

In another thread I have presented a graph, which shows some of such real values s for complex t: but my "tracing" idea is not yet implemented. For a another quick view into it I asked someone to help me to plot a graph for complex t in the complex unit square indicating the contour where t^(1/t) is purely real-valued.
The plot shows this (in principle, it is far too granular!) by colors for imag(s) where t = x + I*y. Where the color is black, purely real values occur for s, because black means imag(s)~0. The real value of s is not visible from that plot, but I showed in my other graph, that it ranges over a wide range up to sinh(10) and above for some complex t, taken from coordinates in that unit square.

Here is the rough plot (only 129 x 64 points in the right half of the unit-square. The black regions should collapse into black lines only and their curved forms resemble roughly the bold red interpolation-lines in my other graph.
The granularity of this plot is far too bad; unfortunately I don't own mathematica&co, and wouldn't feel good when asking the correspondent for more than one or the other favor.

Gottfried

[update] plot removed; improvement see second-next post