How to find the closed form of $\sum_{k=2}^{\infty}{\lambda(k)-1\over k}?$ Given that:
$$\sum_{k=2}^{\infty}{\lambda(k)-1\over k}\tag1$$
Where $\lambda(k)$ is Dirichlet Lambda Function
We are seeking to determine the closed form $(1)$ and came close to estimates it to $1-{\frac12}\left(\gamma+\ln{\pi}\right)$.
where $\gamma=0.5772156...$ is Euler-Mascheroni Constant
How can we evalauate the exact closed form for $(1)$?

Another approach is to use the ordinary generating function of the Riemann zeta function, namely $$\sum_{n=2}^{\infty} \zeta(n) x^{n-1} = - \gamma - \psi (1-x), \quad |x| <1. $$
(This is just the Maclaurin series of the digamma function $\psi(1-x)$.)
Integrating both sides of the generating function, we get
$$\sum_{n=2}^{\infty} \frac{\zeta(n)}{n} \, x^{n} = -\gamma x + \log \Gamma(1-x) + C, $$ where the constant of integration must be zero since the value of the series is $0$ when $x=0$.
Therefore,
$$\begin{align} &\sum_{n=2}^{\infty} \frac{\lambda(n)-1}{n} \\ &= \sum_{n=2}^{\infty} \left(\frac{\zeta(n)}{n} - \frac{1}{n} \right) - \sum_{n=2}^{\infty} \frac{\zeta(n)}{2^{n}n} \tag{1}\\&= \lim_{x \to 1^{-}}\sum_{n=2}^{\infty} \left(\frac{\zeta(n)}{n} - \frac{1}{n} \right)x^{n} - \sum_{n=2}^{\infty} \frac{\zeta(n)}{2^{n}n} \tag{2}  \\ &=\lim_{x \to 1^{-}}\left({\color{red}{\sum_{n=2}^{\infty} \frac{\zeta(n)}{n}x^{n}}} - {\color{green}{\sum_{n=2}^{\infty}\frac{x^{n}}{n}}} \right) - \sum_{n=2}^{\infty} \frac{\zeta(n)}{2^{n}n} \\ &= \lim_{x \to 1^{-}} \big({\color{red}{\log \Gamma(1-x)- \gamma x}}  {\color{green}{ + \ln(1-x) + 1}} \big) + \frac{\gamma}{2} - \log \Gamma \left(1-\frac{1}{2} \right) \\ &=\lim_{x \to 1^{-}} \big(\log \Gamma(1-x) + \ln(1-x) \big)+1  -\frac{\gamma}{2}  - \frac{1}{2} \,  \log \pi  \\ &= \lim_{x \to 1^{-}} \log \big(\Gamma(1-x)\cdot (1-x) \big)+1 -\frac{\gamma}{2} - \frac{1}{2} \,  \log \pi \\  &= \lim_{x \to 1^{-}} \log \Gamma(2-x) +1  -\frac{\gamma}{2} - \frac{1}{2} \,  \log \pi  \\ &=0 +1 -\frac{\gamma}{2}  - \frac{1}{2} \,  \log \pi \\ &= 1 - \frac{1}{2} \left(\gamma + \log \pi \right) \end{align}$$

$(1)$ $\lambda(s) = (1-2^{-s}) \zeta(s)$
$(2)$ Abel's theorem
