A nice relationship between $\zeta$, $\pi$ and $e$

The left-hand side is $$\sum_{n\ge 1}\frac{1}{\Gamma (2n)n(2n+1)4^n}\int_0^\infty\frac{x^{2n-1} dx}{e^x-1}=\int_0^\infty\frac{dx}{e^x-1}\sum_{n\ge 1}\frac{(x/2)^{2n-1}}{(2n+1)!}\\=\int_0^\infty\frac{dx}{e^x-1}\frac{\sinh\tfrac{x}{2}-\tfrac{x}{2}}{(\tfrac{x}{2})^2}.$$The rest is an exercise in complex analysis.

An approach that does not require complex analysis:

$$ \sum_{n=1}^\infty \frac{\zeta(2n)}{n(2n+1)4^n} =\sum_{n=1}^\infty \frac{\zeta(2n)}{n4^n}-2\sum_{n=1}^\infty \frac{\zeta(2n)}{(2n+1)4^n}=S_1-2S_2 $$

To calculate $S_1$: $$\begin{align} S_1=\sum_{n=1}^\infty \frac{\zeta(2n)}{n4^n} &=\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{1}{n(4k^2)^n}\\ &=\sum_{k=1}^\infty \sum_{n=1}^\infty \frac{1}{n(4k^2)^n}\\ &=-\sum_{k=1}^\infty \ln\bigg(1-\frac{1}{4k^2}\bigg)\\ &=-\ln \prod_{k=1}^\infty \frac{(2k+1)(2k-1)}{(2k)^2}\\ &=\ln(\pi/2) \end{align}$$ using the Wallis Product.

You may calculate $S_2$ similarly.

The identities $$\begin{align}\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z^{2k}&=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)&&(\lvert z \rvert < 1)\\ \sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}&=\frac{1}{2}-\frac {1}{2}\ln 2\end{align}$$ (DLMF 25.8.8, 25.8.9) follow from the more basic identity $$\begin{align} \sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^{k}&=-\gamma z+\ln\Gamma\left (1-z\right)&&(\lvert z \rvert < 1) \end{align}$$ (DLMF 25.8.7), which in turn follows from the product formula for the Gamma function (DLMF 5.8.2): $$\frac{1}{\Gamma\left(z\right)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{% z}{k}\right)\mathrm{e}^{-z/k}\text{.}$$