The $L$-polynomial of hyperelliptic function fields and its applications

Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T, \sqrt[\ell]{T^2+aT+b})$ with $a, b \in \F_q$ and $a^2-4b \ne 0$. Using our formula, we obtain the explicit closed formula for the class number of $K$, where $m$ is even or $m=\frac{\ell-1}{2}$.As an application, we compute the average class numbers for hyperelliptic function fields with genus up to $3$.