The answer is yes.
First, let me point out that in general, in ZFC we are not able to refer to the notion of first-order-definable-in-$L$, since definability is not expressible. But in your case, we have $0^\#$, from which we are able to define a truth predicate for first-order truth in $L$, and so your question can be formulated.
For each natural number $n$, the least $\Sigma_n$-correct cardinal $\kappa_n$ is definable in $L$, meaning $L_{\kappa_n}\prec_{\Sigma_n}L$, since we can express this property using a $\Sigma_n$ truth predicate. Further, any $\Sigma_n$ definable ordinal will be bounded by $\kappa_n$. Thus, $\kappa=\sup_n\kappa_n$.
But the union of an increasingly elementary chain is elementary, so $L_\kappa\prec L$ and thus $\kappa$ is a Silver indiscernible.