We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$.
Using the notations of Dirichlet series given in subsection I.E, express, for values of the real $s$ to be specified, $L_{\mathbf{f}}(s)$ in terms of $w$ and $L_{\mathbf{1}}(s)$.