Special Values of p-adic L-functions and Iwasawa λ-invariants of Dirichlet Characters

We study the Iwasawa λ-invariant of Dirichlet characters χ for odd primes p. From the behaviour of the p-adic L-function at s=0, s=1 and s=1−p, we derive easily computable novel criteria to distinguish between the cases λ=0, λ=1, λ=2 and λ≥3. In particular, we look at the case where the p-adic L-function and the Iwasawa power series have a trivial (exceptional) zero. We give a condition for λp(χ)>1 using the p-adic valuation of the Bernoulli number Bp,χω−1 or alternatively Lp(1,χ). We also provide a similar condition for λp(χ)>2. The formulas are used to obtain numerical data on λ-invariants, where either a prime or a Dirichlet character is fixed. The data is consistent with our earlier conjecture on the distribution of λ-invariants. Now the case of a trivial zero is included, for which the distribution of λ-values is shifted by 1.