Let $\langle L_\alpha \rangle$ denote the hierarchy of constructible sets namely $$L_0 = \emptyset$$ $$L_{\alpha+1} = \text{def}(L_\alpha)$$ $$L_{\gamma} = \bigcup_{\alpha<\gamma}{L_\alpha}$$ for $\gamma$ being limit ordinals and $$L = \bigcup_{\alpha \in \text{Ord}}{L_\alpha}$$ be the Godel constructible universe. It is well known that the ordinals are all in $L$.

In $L$, one can also construct this hierarchy and we call it the relativized constructible hierarchy, denoted by $\langle L_\alpha^L \rangle$.

It is easy to see that $\alpha^L = \alpha$ for any ordinal $\alpha$ and $L^L = L$ (i.e. $V=L$ holds in $L$). I want to ask whether it is true that $$L_\alpha^L = L_\alpha.$$