  Let $\mathbb{A}$ be the adele ring of a totally real algebraic number field
$F$. We push forward an explicit computation of a relative trace formula for
periods of automorphic forms along a split torus in $GL(2)$ from a square free
level case done by Masao Tsuzuki, to an arbitrary level case. By using a
relative trace formula, we study central values of automorphic $L$-functions
for cuspidal automorphic representations of $GL(2, \mathbb{A})$ corresponding
to Maass forms with arbitrary level.
