The Hecke $L$-function $H_j(s)$ attached to the $j$th Maass form for the full modular group is estimated in the mean square over a spectral interval for $s=\frac12+it$. As a corollary, we obtain the estimate $H_j(\frac12+it)\ll t^{1/3+\varepsilon}$ for $t\gg\kappa_j^{3/2}$, where $1/4+\kappa_j^2$ is the respective $j$th eigenvalue of the hyperbolic Laplacian. This extends a result due to T. Meurman.