  We show that for a positive proportion of Laplace eigenvalues $\lambda_j$ the
associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary
precision any target function $f(s)$ on a closed disc with center in $3/4$ and
radius $r<1/4$. The main ingredients in the proof are the spectral large sieve
of Deshouillers-Iwaniec and Sarnak's equidistribution theorem for Hecke
eigenvalues.
