Is there a closed form for the following infinite product?
The beautiful idea of Raymond Manzoni can actually be made rigorous. Consider a finite product ∏Ln=1 and take its logarithm. After using duplication formula for the gamma function and telescoping, it simplifies to the following:
This is an exact relation, valid for any L. Now it suffices to use Stirling,
So the answer is indeed 8√πe2.