sequences and series - $sum_{ngeq 1}frac{(-1)^n ln n}{n}$

How can we compute the series $\displaystyle \sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$?

I know it is $\eta '(1)$ , where $\eta$ is the $\eta$ Dirichlet Function , i know its value. But I don't know how to compute it.

An approach I tried is to expand the series, then gather together the odd and the even terms , use $\zeta$ (Riemann's function) and that's all. Then no idea.

Any ideas are welcome.