# more morning speculation

since
$\int_{0}^{\infty} \frac{dx}{\Gamma(x^{3})} \approx 0.753065231886\ldots$

and
$\int_{0}^{\infty} \frac{dx}{\Gamma(x^{2})} \approx 1.165261151780\ldots$, then there is some number $\mu$ between three and two such that

$\int_{0}^{\infty} \frac{dx}{\Gamma(x^{\mu})} = 1$