Given the linear integral of Fredholm type with symmetric kernel for $f:\mathbb{R}_+\to \mathbb{R}$ by \begin{equation} f(x)=\frac{1}{1+x}-\lambda \int_0^\infty \frac{dt\,f(t)}{1+x+t}. \end{equation} The radius of convergence in $\lambda$ should be $\frac{1}{\pi}$.
What is the solution? I'm in particular interested in the point $f(0)$, which has in the $\lambda$ expansion values of the Riemann Zeta function at even positive integers.