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Hereafter, $\ds{\Psi}$ and $\ds{\gamma}$ are the Digamma Function and the Euler-Mascheroni Constant, respectively.
With the well known identity
$\ds{\left.\vphantom{\Large A}\Psi\pars{1 + z}\,\right\vert_{\ \verts{z}\ <\ 1} =
-\gamma + \sum_{n = 2}^{\infty}\pars{-1}^{n}\,\zeta\pars{n}z^{n - 1}}$, we can show that
\begin{align}
&\fbox{$\ds{\ \sum_{n = 1}^{\infty}\zeta\pars{2n}z^{2n}\ }$} =
\half\,z\bracks{\Psi\pars{1 + z} - \Psi\pars{1 - z}} =
\half\,z\bracks{\Psi\pars{z} + {1 \over z} - \Psi\pars{1 - z}}
\\[3mm] = &\ \fbox{$\ds{\ \half - {\pi \over 2}\,z\cot\pars{\pi z}\ }$}
\quad\imp\quad
\left\lbrace\begin{array}{lcl}
\ds{\color{#f00}{\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over 4^{n}}} = \color{#f00}{\half}} & \mbox{with} &
\ds{z = \half}
\\[2mm]
\ds{\color{#f00}{\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over 16^{n}}} = \color{#f00}{{4 - \pi \over 8}}} & \mbox{with} &
\ds{z = {1 \over 4}}
\end{array}\right.
\end{align}
Moreover,
\begin{align}
\int_{0}^{z}2\sum_{n = 1}^{\infty}\zeta\pars{2n}x^{2n - 1}\,\dd x & =
\int_{0}^{z}\bracks{{1 \over x} - \pi\cot\pars{\pi x}}\,\dd x
\\[3mm] \imp
\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over n}\,z^{2n} & =
\ln\pars{\pi z} - \ln\pars{\sin\pars{\pi z}}
\\[3mm] \imp &
\left\lbrace\begin{array}{lcl}
\ds{\color{#f00}{\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over n\, 4^{n}}}
= \color{#f00}{\ln\pars{\pi} - \ln\pars{2}}}
& \mbox{with} &
\ds{z = \half}
\\[2mm]
\ds{\color{#f00}{\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over n\, 16^{n}}} =
\color{#f00}{\ln\pars{\pi} - {3 \over 2}\,\ln\pars{2}}} & \mbox{with} &
\ds{z = {1 \over 4}}
\end{array}\right.
\end{align}