A good part of this talk will be an introduction to the
topic. We will discuss the p-adic analogues of the Riemann zeta function
and Dirichlet L-functions which were constructed in the early 1960s by
Kubota and Leopoldt. We also want to discuss a new proof of an old formula
proved by Bruce Ferrero and myself in the 1970s. That formula gives
the value of the derivative for a Kubota-Leopoldt p-adic L-function
at s=0 when the function itself vanishes at that point. The new proof
is a joint project with Benjamin Lundell and Shaowei Zhang and relies on
studying a certain anaytic function of two p-adic variables.
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