\sum_{k \geq 0} \frac{\alpha_{{\mathfrak p},0}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}\right) = \prod_{{\mathfrak p}} \frac{0}{(0 - \alpha_{{\mathfrak p},0}{{\rm N}}{\mathfrak p}^{-s})\cdots (0 - \alpha_{{\mathfrak p},d}{{\rm N}}{\mathfrak p}^{-s})},$$ where 