Special Values of P-adic L-functions and Iwasawa λ-invariants of Dirichlet characters

We investigate the Iwasawa λ-invariant λp(χ) of the p-adic L-function associated with a Dirichlet character χ for odd primes p. Using special values of the p-adic L-function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases λ=0, λ=1, λ=2, and λ⩾3. In particular, we study the case when the p-adic L-function vanishes at s=0. Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for λp(χ)>1 and λp(χ)>2. Furthermore, we extend the methods of Ernvall-Metsänkylä and of Dummit et al. to calculate the λ-invariant. This is achieved by twisting χ by characters ψ of p-power order and using the values of the p-adic L-function at s=2-p,⋯,0. Additionally, we utilize the value at s=1 to compute λp(χ). Finally, these results are applied to generate numerical data on the distribution of λ-invariants, when either the prime p or the Dirichlet character χ is fixed.