If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\longrightarrow L$, we have a (well-founded) ultrapower $Ult_U(L)$ for a weakly amenable $L$-ultrafilter $U$, but is it iterable?, I mean, can we generate the whole sequence of iterated ultrapowers $\{Ult_U^\alpha(L)\}_{\alpha\in On}$?

I know that this is equivalent to the existence of the sequence $\{Ult_U^\alpha(L)\}_{\alpha\in \omega_1}$, and that a sufficient condition is the $L$-ultrafilter being (externaly) countably complete, which is obviously satisfied for $V$, but maybe not for $L$.