  For a weight two modular form and a good prime $p$, we construct a vector of
Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use
this vector to put the $p$-adic analogues of the conjectures of Birch and
Swinnerton-Dyer for ordinary [MTT] and supersingular [BPR] primes on one
footing. Looking at $L_p^\sharp$ and $L_p^\flat$ individually leads to a
stronger conjecture containing an extra zero phenomenon.
  We also give an explicit upper bound for the analytic rank in the cyclotomic
direction and an asymptotic formula for the $p$-part of the analytic size of
the \v{S}afarevi\v{c}-Tate group in terms of the Iwasawa invariants of
$L_p^\sharp$ and $L_p^\flat$. A very puzzling phenomenon occurs in the
corresponding formulas for modular forms. When $p$ is supersingular, we prove
that the two classical $p$-adic $L$-functions ([AV75],[VI76]) have finitely
many common zeros, as conjectured by Greenberg.
