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An explicit transformation for the series$\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1},$ Re$(y)>0$,which takes $y$ to $1/y$, is obtained for the first time. This seriestransforms into a series containing $\psi_1(z)$, the derivative of Deninger'sfunction $R(z)$. In the course of obtaining the transformation, new importantproperties of $\psi_1(z)$ are derived, as is a new representation for thesecond derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$evaluated at $b=1$. Our transformation readily gives the complete asymptoticexpansion of $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1}$as $y\to0$. An application of the latter is that it gives the asymptoticexpansion of $\displaystyle\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)e^{-\deltat}\, dt$ as $\delta\to0$.

Lambert series of logarithm, the derivative of Deninger's function  $R(z)$ and a mean value theorem for  $\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)$