Let $k=mathbb{F}_q(T)$ be a rational function field over the finite field $mathbb{F}_q$, where q is a power of an odd prime number, and $mathbb{A}=mathbb{F}_q[T]$. Let ${gamma}$ be a generator of $mathbb{F}^*_q$. Let $mathcal{H}_n$ be the subset of $mathbb{A}$ consisting of monic square-free polynomials of degree n. In this paper we obtain an asymptotic formula for the mean value of $L(1,{chi}_{gamma}{small{D}})$ and calculate the average value of the ideal class number $h_{gamma}small{D}$ when the average is taken over $D{in}mathcal{H}_{2g+2}$.