Conjecture $\sum_{n=0}^\infty a_n= \frac{1}{2}-\frac{7 \zeta(3)}{2 \pi^2}$

For $\Re(s)> 0$, let $F(s)$ be defined as$$ \small F(s) = \sum_{n=0}^\infty (-1)^n\left[(n+1)^{1-s} + \frac 1 2 (n+1)^{-s} + \color{red}{(n+1)(n+2)}\left[(n+1)^{-s}\ln(n+1) - \color{red}{(n+2)^{-s}\ln(n+2)}\right]\right]. $$ Then $F(s)$ is analytic and $\displaystyle S= \lim_{\substack{s\to 0\\\Re(s)>0}}F(s)$ holds. Assume $\Re(s)>3$ for a moment so that each term in the summand is absolutely summable. Then by making $n+1\mapsto n$ to the red-colored term, we get \begin{align*} F(s) =& \sum_{n=0}^\infty (-1)^n\left[(n+1)^{1-s} + \frac 1 2 (n+1)^{-s} + 2(n+1)^{2-s} \ln(n+1)\right]\\ =&\eta(s-1) +\frac 1 2 \eta(s) -2\eta'(s-2) ,\qquad \Re(s)>3 \end{align*} where $\eta(s)$ is the Dirichlet's eta function. By analytic continuation, this should also hold for all $\Re(s)>0$ and it follows $$ S = \lim_{\substack{s\to 0\\\Re(s)>0}}F(s) = \eta(-1) + \frac 1 2 \eta(0)-2\eta'(-2)=\frac 1 2 - \frac{7\zeta(3)}{2\pi^2} $$ where the result can be derived from the functional equations satisfied by $\eta(s)$ and $\zeta(s)$, i.e. $$ \eta(s) = (1-2^{1-s})\zeta(s),\qquad \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s). $$

Splitting up the positive and negative terms gives

$$S=\sum_{n=1}^\infty-1+2n(2n-1)\ln\left(1-\frac1{2n}\right)-2n(2n+1)\ln\left(1-\frac1{2n+1}\right)$$

Since $\displaystyle\ln\left(1-\frac1n\right)=\ln(n-1)-\ln(n)=-\int_0^1\frac1{x+n-1}~\mathrm dx$ we provide the more suitable form

$$S=\int_0^1\sum_{n=1}^\infty-1+\frac{2n(2n+1)}{x+2n}-\frac{2n(2n-1)}{x+2n-1}~\mathrm dx$$

The summand can be managed with some long division and the digamma function, reducing this down to

$$S=\int_0^1x(x-1)\sum_{n=1}^\infty\frac{(-1)^n}{x+n}~\mathrm dx\\=\frac12\int_0^1x(1-x)\left[\psi^{(0)}\left(\frac{x+2}2\right)-\psi^{(0)}\left(\frac{x+1}2\right)\right]~\mathrm dx$$

Substituting $x\mapsto1-x$ in the second digamma function and using the reflection formula gives us

$$S=\frac12\int_0^1x(1-x)\left[\frac2x-\pi\cot\left(\frac{\pi x}2\right)\right]~\mathrm dx$$

which reduces to a few polylogarithms and the rest is plugging and chugging.