Try looking in the book by Bellaiche and Chenevier, "Families of Galois representations and higher rank Selmer groups" (published in Asterisque, or available from the Arxiv or Chenevier's web page)here). They give a nice description (Conjecture 5.1.3 in the arxiv versionon the web) of the Bloch-Kato conjecture for the order of vanishing of the L-function for a general "geometric" Galois representation.
They don't state a conjecture for the exact leading term; I don't know what this would look like in general.
