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

[Ivars]

Question:

Do there exist any other roots z and y for equations:

h(z^y) = e^(i*pi/2+-I*2pi*k) ? for all k, or just for some single special k?

So we have in fact 3 pairs of z and y for which solutions exist:

z=e, y=pi/2 , z^y= e^pi/2 h(e^pi/2)= +-i
z=i, y=-i, z^y = i^-i= e^pi/2 h (i^-i)=+-i
z=-i, y=i, z^y = -i^i = e^pi/2 h(i^-i) = +-i

it also follows from formula h(1/(i^i)) = 1/i = -i
and h(1/(-i^-i)) = 1/-i = i.

so this formula h(1/z^z) =1/z must be valid not only for integer and real x, but also for imaginary and maybe complex z.

I find this very intriguing. But still do not know how to extend it beyond first 2 branches of W.

By the way, Gotfrieds curve which crosses imaginary axis at +- i seems to have minumal REAL value at

z= 25*pi=78,5398163397448

and is equal to h(25*pi). More exactly,

h(25pi) = -0,170356724523878+- I* 0,44394520769755.

I could not prove it, but it seems that:

h(25pi)^1/h(25pi) = 25 pi

h(25pi)^(-1/h(25pi)) = 1/(25*pi) =0,012732395

I wonder if other smaller curves on Gottfrieds initial plot ( where Im(h(z) = 0) might have crossing points with imaginary axis and real minimums which have also "nice" z values. If so, the whole curve might be a reasonably nice function- or is it already well known?