Solving Local Minimum of Gamma Function with Integral Calculation

[Doom of Doom]

My physics professor introduce us to the gamma function a few weeks ago, and I've been thinking about it quite a bit. Is there some sort of significance to the fact that the Gamma function has a local minimum value at about n=1.462?

I worked out that the derivative of the gamma function can be shown to be

Integral t^(n-1)*e^(-t)*ln(t) dtand found the zero of that function using my calculator.

 

[Doom of Doom]

A better value for this is n=1.461632

And gamma(1.461632)=.885603