Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say $\lambda=1/4+t^2>1/4$, so $t \in \mathbb{R}_{>0}$.

A resource where somebody has computed the lowest eigenvalue for this case, like is done in LMFDB for the weight zero case, would be optimal. We're not looking to go through the guts of analogous methods to compute this for ourselves, if we can help it.