Evaluating $\int_{0}^{1} \int_{0}^{1}\log (\Gamma(s,x)) \,ds \,dx $

Question

The following result for the logarithm integral of the gamma function is well known

$$\int_{0}^{1}\log(\Gamma(x)) \,dx = \dfrac{1}{2}\log 2 \pi \tag {1}$$

I want to evaluate a two-variable similar integral but with the incomplete gamma function $\Gamma(s,x)$ given by

$$I = \int_{0}^{1}\int_{0}^{1}\log(\Gamma(s,x)) \,ds\,dx \tag {2}$$

WolframAlpha gives $I \approx -0.515942$ but I have no idea how to prove this result.

Is there some "pretty" result for $(2)$ like that one of the integral $(1)$?

Thanks for any help!