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 Sarnaks equidistribution theorem for Hecke eigenvalues.
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