Define a Leibniz series as follows, \begin{eqnarray*} L(x) & = & \sum_{k=1}^{\infty}(-1)^{k}e^{-kx}\ln k,\ \ x>0 \end{eqnarray*}
I have two questions: (I) Is there an analytical form for $L(x)$? (II) Does the analytic continuation for $L(x)$ from region $x>0$ to region $x<0$ exist?