Functional Square Root

[tommy1729]

If the fixed point of f is any L not 1, we'll have L*e=exp(L) and exp(f'(L))=exp(L)f'(L)=e*L*f'(L), again we have to find a solution of exp(x)=e*L*x whose absolute value remains smaller than 1. As I computed,
\[L=-W_1(-\tfrac{1}{e})\approx{3.08884-7.46149i}, s=f'(L)=-W_0\bigg(\frac{1}{eW_1(-\tfrac{1}{e})}\bigg)\approx{0.0158263+0.0434912i}\]

The real part of L is close to the Tetra-Euler Number. 

my intuition says that is no coincidence ... but no detailed rational ideas yet.