$\pi~$ expanded in terms of the Euler-Mascheroni Constant $\gamma$

Question

Is there a known expansion of $\pi$ as a function of the Euler-Mascheroni $\gamma$? As in, $$f(\gamma)=\pi h(\pi,\gamma)$$ where $\gamma$ appears alone with only rational arguments like $f(1/2,\gamma)$, and $h(\pi,\gamma)$ is a single, or finite closed form expression of the two constants, like $\pi \zeta(\pi\gamma)$, not considering (trivial) change of variables in integrals as in here. This page seems to have only $$h(\pi,\gamma)\gamma=f(\pi)$$ the inverse of what I ask.