The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in $L$.
Since $On⊂L⊆V$, properties of ordinals that depend on the absence of a function or other structure (i.e. $\Pi_1^{ZF}$ formulas) are preserved when going down from $V$ to $L$. Hence initial ordinals of cardinals remain initial in L. Regular ordinals remain regular in $L$. Weak limit cardinals become strong limit cardinals in $L$ because the generalized continuum hypothesis holds in $L$. Weakly inaccessible cardinals become strongly inaccessible. Weakly Mahlo cardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinal properties) will be retained in $L$.
Let $\kappa$ be a Mahlo cardinal in $V$, i.e. let $\kappa$ be inaccessible such that $S:= \{ \alpha \in \kappa \mid \alpha \text{ is regular} \}$ is stationary in $\kappa$.
First, note that $L \models \kappa \text{ is inaccessible}$. Indeed, if $L \models \kappa \text{ is not a cardinal}$, then there is some $\mu < \kappa$ and some $f \in L$ such that $L \models f \colon \mu \to \kappa \text{ is surjective}$. However, this is a $\Sigma_{0}$ property and hence, in $V$, $f \colon \mu \to \kappa$ is surjective. Contradiction. Since $\kappa > \omega$ (as an ordinal), this also yields that $L \models \kappa \text{ is uncountable}$. Repeating this argument with cofinal $f \colon \mu \to \kappa$ yields that $L \models \kappa \text{ is regular}$.
Since $L \models \operatorname{GCH}$, it now suffices to prove that $L \models \kappa \text{ is a limit cardinal}$. This is trivial, because for any ordinal $\gamma < \kappa$, we have that $(\gamma^{+})^V < \kappa$ and since cardinals in $V$ are cardinals in $L$, this proves $$L \models \forall \gamma < \kappa \exists \gamma < \mu < \kappa \colon \mu \text{ is a cardinal}.$$
Now let $T := \{ \alpha \in \kappa \mid L \models \alpha \text{ is regular} \}$. By the argument given above, any $\alpha$ that is regular in $V$ is regular in $L$ and hence $S \subseteq T$. Suppose that $L \models \kappa \text{ is not Mahlo}$. Then there is some $C \subseteq \kappa$ such that $L \models C \text{ is club in } \kappa \text{ and } C \cap T = \emptyset$. Being club is a $\Sigma_0$, property and hence $C \subseteq \kappa$ is club in $\kappa$. Since $S \subseteq T$, we have that $C \cap S = \emptyset$ and hence $S$ is not stationary in $V$. This is a contradiction and we therefore must have that $L \models T \text{ is stationary}$.
Thus, $\kappa$ remains Mahlo in $L$.