  We consider a Wigner matrix $A$ with entries tail decaying as $x^{-\alpha}$
with $2<\alpha<4$ for large $x$ and study fluctuations of linear statistics
$N^{-1}\operatorname{Tr}\varphi(A)$. The behavior of such fluctuations has been
understood for both heavy-tailed matrices (i.e. $\alpha < 2$) and light-tailed
matrices (i.e. $\alpha > 4$). This paper fills in the gap of understanding for
$2<\alpha<4$. We find that while linear spectral statistics for heavy-tailed
matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed
matrices have fluctuations of order $N^{-1}$, the linear spectral statistics
for half-heavy-tailed matrices exhibit an intermediate $\alpha$-dependent order
of $N^{-\alpha/4}$.
