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Generally the values of L-functions at positive integers are transcendental numbers, or are numbers believed to be transcendental. There isn't a natural way to embed these iside an algebraic closure of $\mathbb{}_p$, so I cannot see how there's a sensible notion of interpolation.
This kind of numerology generalizes to general finite Abelian $p$-groups via the formalism of Hall polynomials: en.wikipedia.org/wiki/Hall_polynomial .
