6. The value at s = 1
Throughout, p is an odd prime and \theta is a non-trivial Dirichlet
character, written in the usual form \theta = \chi\eta, where \chi has
conductor p^n and \eta has conductor D prime to p. We write
N = Dp^n for the conductor of \theta, and let \varepsilon_N denote a
fixed primitive N-th root of unity. We write G(\theta) for the Gauss sum
attached to \theta.
The interpolation theorem identifies, for every integer k > 0, the integral
\int_{\Zpx}\chi(x)x^k\,d\zeta_\eta with the classical L-value
L(\theta,1-k). These k>0 lie in the range of interpolation, and there the
link to classical values is explicit; such values are called critical. It is
natural to ask what happens outside this range — in particular at k=0,
i.e. at s=1. A priori such non-critical values need have nothing to do with
classical L-values: indeed L(\theta,1) is transcendental, so it cannot be
viewed as a p-adic number in any natural way. Nevertheless there is a formula
for the p-adic L-function at s=1 strikingly parallel to its classical
analogue. The p-adic formula is due to Leopoldt; it can be used to prove a
p-adic analogue of the analytic class number formula, and is the simplest
instance of the p-adic Beilinson / Perrin-Riou conjectures, which describe
non-critical special values of p-adic L-functions in arithmetic terms.