Actually, *p*-adic *L*-functions are expected to satisfy functional equations compatible with the classical ones. For *M* an ordinary motive, Coates and Perrin-Riou conjectured the interpolation property at critical integers and the expected functional equation in some papers in the early nineties (see for example [this][1]). In particular, the Kubota-Leopoldt *p*-adic *L*-functions interpolate *all* critical values of the classical Dirichlet *L*-functions (up to a period and a multiple). For modular forms, Mazur-Tate-Teitelbaum, in their 1986 paper) prove a *p*-adic functional equation in section 17. In fact, the two-variable *p*-adic *L*-function of an ordinary family of modular forms satisfies a two-variable functional equation interpolating the one-variable functional equation at each weight (see for example Greenberg-Stevens' inventiones paper) (I'd post more mathscinet links but it appears to be down...).

As for the values of the *p*-adic *L*-function at non-critical integers, that's much more mysterious. Rubin has a computation outside of the critical points for a CM elliptic curve in section 3.3 of his paper in the "*p*-adic monodromy and BSD" proceedings. I think I've seen other cases, but generally it takes a lot of effort, I think.

(Also, regarding Iwasawa theory's concern with values of *L*-functions, it is true that the Main Conjecture is only an equality of ideals in some power series ring, but one can still hope to construct *p*-adic *L*-functions on the analytic side that do a nice job at interpolating, say up to a *p*-adic unit.)

  [1]: http://www.ams.org/mathscinet-getitem?mr=1097608