Yes, the parameter-free version of $L$ gives rise to the same constructible universe $L$. You will still get all of $L$ this way, but it will come more slowly.
The reason is that at stage $\alpha+1$, you in effect have $\alpha$ as a parameter, since this is definable as the largest ordinal. So you can use $\alpha$ as a parameter. Every finite sequence of ordinals is coded by a single ordinal, and so in this way you can have any finitely many ordinal parameters, and this is enough to pick out any object in the resulting hierarchy. So eventually you will get every set in $L$.