Interesting value for W, h involving phi,Omega?

[Ivars]

Next
I tried to find h(z) = h((I/3)^(3/I)) = h( exp(3pi/2)/exp(-I*3*ln3))

ln ( exp(3pi/2)/exp(-I*2*ln2)) = 3pi/2+ I*3*ln3

-ln(z) = - 3pi/2- I*3*ln3

so as W(-3pi/2-I*3*ln3) = ln3 - I*(pi/2)

Than h(z) = (-ln3+I*(pi/2))/ (-3pi/2 - i*3*ln3)

This can be brought to :

h(z) = I/3 ( i guess with plus sign, not sure)

For h(i/4)^(4/i)) we have :

W(-2pi-I*4*ln4) = ln4 -I*(pi/2)

So most likely:

W(-n*pi/2 -I*n*ln n) = ln n -n* I*(pi/2) n>1