Interesting value for W, h involving phi,Omega?

[Ivars]

I was studying the graph of selfroot of Lambert function:

W(x)^(1/W(x)).

It has maximum value at x=(e*(e^e)) and is e^(1/e) ; so

W(e*(e^e)) = e

Numerically,

W(41,1935556747..)^(1/(W(41,1935556747)= 1,444667861

I multplied e*(e^e)* Omega constant =

41,193556747..*0,567143...=23,36263675...

On other hand, I took logarithm of (e*(e^e))

ln (e*(e^e)) = 1+e = 3,718281828.....

I multiplied it with Pi :

3,718281828.....* 3,141592..= 11,68132688

And I multiplied this with 2:

11,68132688..*2 = 23,362653...

So:

pi = approx((e*(e^e)*Omega)/(2*(e+1)))

Since e=Omega^(-1/Omega), its just an approximation containing 2 and Omega.

This approximation seems to be good for 5 decimals. I wonder why and can it be improved.