An intriguing relation between two alternating series

This is another intriguing formula coming from Ramanujan's letter to G. H. Hardy dated 16th Jan 1913 $$\frac{\log 1}{\sqrt{1}} - \frac{\log 3}{\sqrt{3}} + \frac{\log 5}{\sqrt{5}} - \cdots = \left(\frac{\pi}{4} - \frac{\gamma}{2} + \frac{\log 2\pi}{2}\right)\left(\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{5}} - \cdots\right)$$ where $$\gamma = \lim_{n \to \infty}\left(\frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n} - \log n\right)$$ is the Euler's constant.

This seems really a different kind of series dealing with $\sqrt{n}$ in the denominator and is so unlike the usual series for $\log (1 + x)$ or $\arctan x$ or even Abel's function $\sum x^{n}/n^{2}$. Please help me out in proving the above identity. Any reference having a proof would also be of great help.

It comes straight from the functional equation for the Dirichlet beta function $\beta(s)$ (proved by Hardy in a very clever way), since: $$ \frac{d}{ds}\frac{1}{n^s}=-\frac{\log n}{n^s}.$$