Using the notations of Dirichlet series given in subsection I.E, and with $w = \frac{1}{\lambda-1}$ and $\mathbf{f} = (1+w)\delta - w\mathbf{1}$, express, for values of the real $s$ to be specified, $L_{\mathbf{f}}(s)$ in terms of $w$ and $L_{\mathbf{1}}(s)$.