We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the p-adic L-functions of Bertolini, Darmon, and Prasanna attached to elliptic curves E/Q at primes p of good ordinary reduction. Using Iwasawa theory, we then prove, under mild hypotheses, one of the inequalities predicted by the “rank part” of our conjectures, as well as the predicted leading coefficient formula, up to a p-adic unit.Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini and Darmon in 1996 for Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.