How to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$?

Question

I need to find first 32 digits of $\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)$ but wolframalpha's brain is too narrow to contain the result, and I don't know any software and programming to find the result. $\gamma$ is the Euler's constant. Or does it converge to a known number? Please help! Thanks

Added for Bounty. Does the mentioned infinite product have any closed form in terms of known mathematical constants?

Answer 1

I asked Alpha and got
$0.1734054215618562128757376353903311358240418\\ 356990083565526180089819971434621977210396477\\ 9552159698610211689009709...$

with one click on More Digits. I have found that sometimes when a calculation fails, close the tab, open a new one, and try again sometimes works.

Answer 2

Using Pochhammer symbols $$\displaystyle \prod_{n=1}^{\infty} (1-\gamma^n)=(\gamma ;\gamma )_{\infty }$$ A very fast way to compute it (have a look here) is $$(\gamma ;\gamma )_{\infty }=\sum_{k=-\infty}^\infty (-1)^k \,\gamma ^{\frac{3k^2-k}{2} }$$ Consider partial sums $$S_p=\sum_{k=-p}^p (-1)^k \,\gamma ^{\frac{3k^2-k}{2} }$$ and compute for $50$ decimal places $$\left( \begin{array}{cc} 5 & 0.17340542156115797462125386932229340638693672477775 \\ 6 & 0.17340542156185623295922558757637084613978106011987 \\ 7 & 0.17340542156185621287562583984349502239524993758969 \\ 8 & 0.17340542156185621287573763551045706783782602427578 \\ 9 & 0.17340542156185621287573763539033111094663184209447 \\ 10 & 0.17340542156185621287573763539033113582404282775260 \\ 11 & 0.17340542156185621287573763539033113582404183569900 \\ 12 & 0.17340542156185621287573763539033113582404183569901 \\ 13 & 0.17340542156185621287573763539033113582404183569901 \\ 14 & 0.17340542156185621287573763539033113582404183569900 \\ 15 & 0.17340542156185621287573763539033113582404183569900 \end{array} \right)$$ For $32$ decimal places $S_9$ would be fine.