How bad is the Elephant Gambit? I tend to think the Elephant Gambit is dubious, but what is the best path to a White advantage?

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1.e4 e5 2.Nf3 d5


Update: 
 RemcoGerlich's comment of an upcoming book is quite relevant!  Typical analysis is only 4 moves deep claiming White is better (as shown in some answers below).

I had never seen 1.e4 e5 2.Nf3 d5 3.exd5 Bd6 given as "Elephant Gambit Proper" on wikipedia (link in Kortchnoi's answer); The wiki, by-the-way, quotes De Firmian analysis which is pre-computer analysis.
Good question. You are right, the Elephant gambit appears to be a dubious opening. Have you checked the wikipedia page on the Elephant Gambit? It offers good lines to fight for an advantage. But they are two other questions:


Do you like to defend or retain the initiative?


Those playing this opening are not afraid to gambit a pawn for the initiative with 1.e4 e5 2. Nf3 d5 3.exd5 e4. In most of the lines following 4.Qe2 Nf6 5.d3 Be7 or 4.Qe2 Nf6 5.Nc3 Be7 white has a pawn up but black has the initiative. With a precise play and a bit of patience, white will gain an advantage. 


Do you like opening theory?


The issue is that you are going to face this rare opening once or twice in your tournament life and most of the time black will be better prepared (in this rare line). He/she will have more experience with this gambit. So, another way to deal with this gambit will be to go for an unrated variant such as 

1.e4 e5 2. Nf3 d5 3.Nxe5 Bd6 4.d4 dxe4 5.Qe2!? 

[fen "rnbqk1nr/ppp2ppp/3b4/4N3/3Pp3/8/PPP1QPPP/RNB1KB1R w KQkq - 0 1"]


followed by Nc3, Bg5 and 0-0-0...

Those continuations have many advantages: Easy to remember; easy to play, and maybe not the kind of positions black are looking for when playing the Elephant Gambit !