Let \theta be a non-trivial Dirichlet character of conductor N, and let \varepsilon_N be a primitive N-th root of unity.

(i) Classical value at s=1. We have L(\theta,1) = -\frac{1}{G(\theta^{-1})}\sum_{c\in(\Z/N\Z)^\times}\theta^{-1}(c)\,\log\!\big(1-\varepsilon_N^{\,c}\big).

(ii) p-adic value at s=1. We have L_p(\theta,1) = -\big(1-\theta(p)p^{-1}\big)\frac{1}{G(\theta^{-1})}\sum_{c\in(\Z/N\Z)^\times}\theta^{-1}(c)\,\log_p\!\big(1-\varepsilon_N^{\,c}\big).

Part (i) rests on the special-value description of Definition 2.1.3. Part (ii) concerns the value at s=1 of the Definition 4.3.2 p-adic L-function, which lies outside the range of the Theorem 5.1.1, and is computed from the measures Definition 4.1.4.