Is $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$ a rational number?

Question

I would like to learn more about the behavior of the factorial function or Gamma function, so I decided to compute the following integral $$ \int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}\,\mathrm dx. $$

According to Wolfram alpha, its value is approximately $3$:

My question is whether the exact value is rational or not.

Edit: The Motivation of this question is to know more about transcendence degree of the field generated by $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$

Answer 1

This number is somewhow similiar to the (more or less) well-known Fransén-Robinson Constant $F$. The latter is defined by a somewhat analogical integral

$$F:=\int_0^\infty\frac{\mathrm dx}{\Gamma(x)}=2.807~770\dots$$

Note that this one is close to $e=2.718~281...$ since the integral may be approximated by the standard infinite sum for Napier's constant.

However, even though the Fransén-Robinson Constant is listed in a bunch of overviews of mathematical constants there is not much more to say about this constant; and I suspect the same for your given one... It seems to be unclear whether there is an "easy" closed-form expression for the Fransén-Robinson constant in terms of other known constant and the question about irrationality is not even tossed in the room on Wikipedia for example (of course, this is a reasonable question for all mathematical constants).

Using Approach0 I have found four posts here on MSE related to the Fransén-Robinson constant

• Laplace transform involving the gamma function.
• Definite Integral over the Gamma Function
• Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$
• Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$

They may be of help while examining

$$\int_1^\infty\frac{\mathrm dx}{\sqrt{\Gamma(x)}}=2.992~866\dots$$

To be honest: I have doubts that this will lead somewhere.