Sum of reciprocals of eigenvalues of a Sturm-Liouville problem

Let $(\lambda_n)_{n\in\mathbb N}$ be the sequence of eigenvalues of the Sturm–Liouville problem

$$\frac d{dx}\Big(x\frac{dy}{dx}\Big)+\frac\lambda xy=0,\quad1<x<e^{2\pi},\qquad y(1)=0, y(e^{2\pi})=0.$$

Then $\displaystyle\sum_{n=1}^\infty\frac1{\lambda_n}$ is equal to