Average value of the divisor class numbers of real cubic function fields

We compute an asymptotic formula for the divisor class numbers of real cubic function fields K m = k (m 3), where F q is a finite field with q elements, q 1 (mod 3), k:= F q (T) is the rational function field, and m ϵ F q [ T ] is a cube-free polynomial; in this case, the degree of m m is divisible by 3. For computation of our asymptotic formula, we find the average value of |L (s, χ)|2 evaluated at s = 1 when χ goes through the primitive cubic even Dirichlet characters of F q [ T ], where L (s, χ) is the associated Dirichlet L -function.