I found this formula for the Euler-Mascheroni constant $\gamma$.

Just wondering whether such a formula already exists in literature? Also, wanted to know whether there are formulas that converge faster than this?

$$\gamma = \sum_{k = 1}^{\infty} \frac{1}{2^k k} - \sum_{k = 1}^{\infty} \frac{\zeta \left( 2 k + 1 \right)}{2^{2 k} \left( 2 k + 1 \right)} $$

UPDATE: Thanks for your reply quid. I just came across this while doing some calculations with the zeta function. The calculations are a bit too long to be posted, but in short it derives from $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi x))^2}{x^{s+2}}dx$$.