  With the method of moments and the mollification method, we study the central
$L$-values of GL(2) Maass forms of weight $0$ and level $1$ and establish a
positive-proportional nonvanishing result of such values in the aspect of large
spectral parameter in short intervals, which is qualitatively optimal in view
of Weyl's law. As an application of this result and a formula of Katok--Sarnak,
we give a nonvanishing result on the first Fourier coefficients of Maass forms
of weight $\frac{1}{2}$ and level $4$ in the Kohnen plus space.
