Rational $p$-adic zeros of the Leopoldt-Kubota $p$-adic $L$-functions give rise to certain sequences of generalized Bernoulli numbers tending $p$-adically to zero, and conversely. This relationship takes different forms depending on whether the corresponding Iwasawa $\lambda$-invariant is one or greater than one. To understand the relationship better it is useful to consider approximate zeros of those functions.
"Bernoulli numbers and zeros of $p$-adic $L$-functions." Funct. Approx. Comment. Math. 39 (2) 223 - 235, December 2008. https://doi.org/10.7169/facm/1229696573