Eigenvalues of the Laplacian $\Delta$ acting on $L^2 (G/ \Gamma)$, where $G = SL_2 (\mathbb{R})$ and $\Gamma = SL_2 (\mathbb{Z}) < G$ (one can consider more general groups $G$ and take any lattice $\Gamma$ in $G$), or the so called Maass forms. It is known, by Selberg's trace formula and other related results, that such eigenvalues do exist, and we even have theorems describing their asymptotic count, but not a single, concrete example of a Maass form is known, even for this specific choice of $G$ and $\Gamma$. Quoting from Goldfeld's "Automorphic forms and L-functions for the group GL(n,R)":
"Up to now no one has found a single example of a Maass form for $SL_2 (\mathbb{Z})$".