Series representation for Euler-Mascheroni constant

In his 1887 paper Table des valeurs des sommes $S_k = \sum_{1}^\infty n^{-k}$ (Acta Mathematica 10 (1887), 299-302; volume available online), Stieltjes used almost exactly this formula to compute Euler's constant to 33 decimal places. Of course as quid points out you need to know the zeta values to do this, but the main point of this paper was to compute those values, so he was just getting Euler's constant as a corollary. He uses a slight variant of the formula, with $\zeta(2k+1)-1$ in place of $\zeta(2k+1)$ for faster convergence (and a corresponding adjustment in the other term, which becomes $1+\log 2 - \log 3$). He derives the formula by taking the Taylor series expansion of $\log \Gamma(1+x)$ and using it to compute $\log \Gamma(1+1/2) - \log \Gamma(1-1/2)$.

The method used for the recent record computations of Euler--Mascheroni is (a refinement of) a classical algorithm due to Bren-McMillan . This algorithm is $O(n (\log n)^3)$.

Now, for you formula (btw, could you say where/how you found it?), it is not quite clear to me what you are asking.

The series involves $\zeta$ values (not only elementary things)! So, if one where to use this to compute approximations of $\gamma$ one would need all kinds of $\zeta$ values at odd naturals, and early ones to essentailly the same precision as one seeks $\gamma$. And, also from a theoretical this makes unclear what type of expressions you would admit as 'competition'.

To continue on the computation bit, if one would only compute/estimate $\zeta(3)$ naively this would already make this worse then above mentioned method. (One can also compute $\zeta(3)$ faster--in the recent record computations for it also an $O(n (\log n)^3)$ algorithm was used (though it appears to be simpler to compute $\zeta(3)$ than $\gamma$); but this is only $\zeta(3)$. And, it reraises the issue that it is unclear how to treat the $\zeta$ values when interpreting your question.)

An overview on algorithms used to calculate these and related constants to high precision in practise, can also be found on that site.