$\sum_{n=1}^{\infty}(-1)^{n-1}\left({\beta(n)\over n}-\ln{n+1\over n}\right)=\ln\sqrt{2\over \pi}\cdot{2\over \Gamma^2\left({3\over 4}\right)}?$
Let us consider the two sums separately: $$ S_1 = \sum_{n=1}^\infty (-1)^{n-1}\frac{\beta(n)}{n} \\ S_2 = \sum_{n=1}^\infty (-1)^{n-1}\ln\frac{n+1}{n} $$
For $S_1$, let us first consider the related sum, for which $S_1$ is the limiting value $S(1)$: $$ S(x) = \sum_{n=1}^\infty (-1)^{n-1}\frac{\beta(n)x^n}{n} \\ S'(x) = \sum_{n=1}^\infty (-1)^{n-1}\beta(n)x^{n-1} = \sum_{n=1}^\infty (-1)^{n-1}x^{n-1}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^n} \\ = \sum_{k=0}^\infty \frac{(-1)^k}{2k+1} \sum_{n=1}^\infty \left( \frac{-x}{2k+1} \right)^{n-1} = \sum_{k=0}^\infty \frac{1}{2k+1+x} = \frac{1}{2} \Phi \left(-1, 1, \frac{1+x}{2} \right) $$ Where $\Phi$ is the Lerch Transcendent, and $|x|<1$. Now, upon integration, we can recover $S_1$: $$ S_1 - S(0) =\int_0^1 S'(x) dx = \int_0^1 \frac{1}{2} \Phi \left(-1, 1, \frac{1+x}{2} \right)dx = \int_\frac{1}{2}^1 \Phi (-1, 1, x)dx \\ = \lim_{s\to0}\int_\frac{1}{2}^1 \Phi (-1, 1+s, x)dx = \lim_{s\to0} \left[ \frac{-1}{s}\Phi (-1, s, x) \right|_\frac{1}{2}^1 \\ = \lim_{s\to0} \frac{\Phi \left(-1, s, \frac{1}{2}\right) - \Phi(-1, s, 1)}{s} = \lim_{s\to0} \frac{2^s \beta(s) - \eta(s)}{s} \\ =^H \lim_{s\to0} \ \ln(2) 2^s \beta(s) + 2^s \beta'(s) - \eta'(s) = \frac{1}{2} \ln(2) + \ln \frac{\Gamma^2(\frac{1}{4})}{2 \pi \sqrt{2}} - \frac{1}{2} \ln \frac{\pi}{2} = \ln \frac{\sqrt{2 \pi}}{\Gamma^2(\frac{3}{4})} $$ And as $S(0) = 0$, this is all $S_1$. For $S_2$: $$ S_2 = \sum_{n=1}^\infty (-1)^{n-1}\ln\frac{n+1}{n} = \ln \prod_{n=1}^\infty \left( \frac{n+1}{n} \right)^{(-1)^{n-1}} \\ = \lim_{N \to \infty} \ln \prod_{n=1}^N \frac{n^2}{(n - \frac{1}{2})(n + \frac{1}{2})} = \lim_{N \to \infty} \ln\frac{\pi \Gamma^2(N+1)}{2\Gamma(N + \frac{1}{2})\Gamma(N + \frac{3}{2})} = \ln \frac{\pi}{2} $$ The sum therefore equals: $$ S_1-S_2=\ln \frac{2\sqrt2}{\sqrt{\pi} \ \Gamma^2(\frac{3}{4})} $$ as predicted.
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\substack{\ds{\sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\bracks{{\beta\pars{n} \over n} -\ln\pars{n + 1 \over n}} = \ln\pars{\root{2 \over \pi}\,{2 \over \Gamma^{\,2}\pars{3/4}}}:\ {\large ?}.} \\[3mm] \ds{\beta:\texttt{Dirichlet Beta Function}.}}$
Lets \begin{equation} \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\bracks{{\beta\pars{n} \over n} -\ln\pars{n + 1 \over n}} = \mc{S}_{1} - \mc{S}_{2}\,,\quad \left\{\begin{array}{l} \ds{\mc{S}_{1} \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,{\beta\pars{n} \over n}} \\[2mm] \ds{\mc{S}_{2} \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,\ln\pars{n + 1 \over n}} \end{array}\right.\label{1}\tag{1} \end{equation}
$\ds{\Large\mc{S}_{1}:\ ?.}$ With the $\ds{\beta}$ Integral Representation: \begin{align} \mc{S}_{1} & \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,{1 \over n}\ \overbrace{{1 \over \Gamma\pars{n}}\int_{0}^{\infty} {x^{n - 1}\expo{-x} \over 1 + \expo{-2x}}\,\dd x}^{\ds{\beta\pars{n}}}\ =\ -\int_{0}^{\infty} {\bracks{\sum_{n = 1}^{\infty}\pars{-x}^{n}/n!}\expo{-x} \over 1 + \expo{-2x}}\,{\dd x \over x} \\[5mm] & = -\int_{0}^{\infty} {\pars{\expo{-x} - 1}\expo{-x} \over 1 + \expo{-2x}}\,{\dd x \over x} \,\,\,\stackrel{\exp\pars{-2x}\ \mapsto\ x}{=}\,\,\, -\int_{1}^{0}{x - x^{1/2} \over 1 + x}\,{\dd x/\pars{-2x} \over \ln\pars{x}/\pars{-2}} \\[5mm] & = \int_{0}^{1}{x^{-1/2} - 1 \over 1 - x^{2}}\,{x - 1 \over \ln\pars{x}}\,\dd x = \int_{0}^{1}{x^{-1/2} - 1 \over 1 - x^{2}}\ \overbrace{\int_{0}^{1}x^{t}\,\dd t}^{\ds{x - 1 \over \ln\pars{x}}}\ \,\dd x \\[5mm] & = \int_{0}^{1}\int_{0}^{1}{x^{t - 1/2} - x^{t} \over 1 - x^{2}}\,\dd x\,\dd t \,\,\,\stackrel{x^{2}\ \mapsto\ x}{=}\,\,\, {1 \over 2}\int_{0}^{1}\int_{0}^{1}{x^{t/2 - 3/4} - x^{t/2 - 1/2} \over 1 - x}\,\dd x\,\dd t \\[5mm] & = {1 \over 2}\int_{0}^{1}\bracks{% \int_{0}^{1}{1 - x^{t/2 - 1/2} \over 1 - x}\,\dd x - \int_{0}^{1}\int_{0}^{1}{1 - x^{t/2 - 3/4} \over 1 - x}\,\dd x}\dd t \\[5mm] & = {1 \over 2}\int_{0}^{1} \bracks{\Psi\pars{{t \over 2} + {1 \over 2}} - \Psi\pars{{t \over 2} + {1 \over 4}}}\dd t \qquad\pars{~\Psi:\ Digamma\ Function~} \\[5mm] & = \left. \ln\pars{\Gamma\pars{t/2 + 1/2} \over \Gamma\pars{t/2 + 1/4}} \right\vert_{\ 0}^{\ 1} = \ln\pars{{\Gamma\pars{1} \over \Gamma\pars{3/4}}\,{\Gamma\pars{1/4} \over \Gamma\pars{1/2}}} \\[5mm] & = \ln\pars{{1 \over \Gamma^{\,2}\pars{3/4}}\,{\Gamma\pars{3/4}\Gamma\pars{1/4} \over \root{\pi}}}\quad \pars{~\substack{\mbox{Note that}\\[2mm] \ds{\Gamma\pars{1 \over 2} = \root{\pi}\,,\ \Gamma\pars{1} = 1}}~} \\[5mm] & = \ln\pars{{1 \over \Gamma^{\,2}\pars{3/4}\root{\pi}} \,{\pi \over \sin\pars{\pi/4}}}\qquad\pars{~Euler\ Reflection\ Formula~} \\[5mm] & \implies \bbx{\mc{S}_{1} \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,{\beta\pars{n} \over n} = \ln\pars{\root{2\pi} \over \Gamma^{\,2}\pars{3/4}}}\label{2}\tag{2} \end{align}
$\ds{\Large\mc{S}_{2}:\ ?.}$ \begin{align} \mc{S}_{2} & \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,\ln\pars{n + 1 \over n} = \sum_{n = 0}^{\infty}\pars{-1}^{n}\int_{0}^{1}{\dd t \over t + n + 1} = \int_{0}^{1}\sum_{n = 0}^{\infty}{\pars{-1}^{n} \over n + t + 1}\,\dd t \\[5mm] & = {1 \over 2}\int_{0}^{1}\sum_{n = 0}^{\infty} \bracks{{1 \over n + \pars{t + 1}/2} - {1 \over n + t/2 + 1}}\,\dd t \\[5mm] & = {1 \over 2}\int_{0}^{1}\bracks{% \Psi\pars{{t \over 2} + 1} - \Psi\pars{t + 1 \over 2}}\,\dd t = \left.\ln\pars{\Gamma\pars{t/2 + 1} \over \Gamma\pars{t/2 + 1/2}} \right\vert_{\ 0}^{\ 1} \\[5mm] & = \ln\pars{{\Gamma\pars{3/2} \over \Gamma\pars{1}}\,{\Gamma\pars{1/2} \over \Gamma\pars{1}}} = \ln\pars{{1 \over 2}\,\Gamma^{\,2}\pars{1 \over 2}} \qquad\pars{~\Gamma-Recursive Property~} \\[5mm] & \implies \bbx{\mc{S}_{2} \equiv \sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\,\ln\pars{n + 1 \over n} = \ln\pars{\pi \over 2}}\label{3}\tag{3} \end{align}
With \eqref{1}, \eqref{2} and \eqref{3}: $$ \bbx{\sum_{n = 1}^{\infty}\pars{-1}^{n - 1}\bracks{{\beta\pars{n} \over n} -\ln\pars{n + 1 \over n}} = \ln\pars{\root{2 \over \pi}\,{2 \over \Gamma^{\,2}\pars{3/4}}}} $$