Can $\ln(\Gamma(e^s))\ln(\zeta(e^s))$ be expressed as an integral?

Question

Consider the known relation $$ \Gamma(s)\zeta(s)= \displaystyle \int_{0}^{\infty}\dfrac{x^{s-1}}{e^{x}-1}\,dx.$$

Can $$ \ln(\Gamma(e^s))\ln(\zeta(e^s)) $$ be expressed as an integral? Maybe there's some substitution that can be done? I haven't been able to express it as an integral yet, so any hints are welcome.

Answer 1

Things aren't magical. If $\Gamma(s)\zeta(s)$ has a simple expression it is not for nothing.

It doesn't apply to $\ln(\Gamma(e^s))\ln(\zeta(e^s))$.