A famous result by Kunen regarding elementary embeddings states that there is no such embedding from $V$ onto itself which would be non-trivial. It's clear that if $V=L$ then there is also no non-trivial elementary embedding from $L$ onto itself. By similar means we know that no such embedding, if exists, can be seen from "inside $L$".
However, I can't see directly that from Kunen's theorem follows non-existence of embedding from $L$ to $L$ which exists in $V$. Something tells me that such embedding would have to be closed between universes, which would already show the impossibility, but I see no way to prove it.
I, also, am no expert in set theory, so I can't look through proof of theorem to see if it's applicable to $L$.
Thanks in advance!