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Let k = F q (T ) be a rational function field over the finite field F q , where q is a power of an odd prime number, and A = F q [T ]. Let γ be a generator of F * q . Let H n be the subset of 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, χ γD ) and calculate the average value of the ideal class number h γD when the average is taken over D ∈ H 2g+2 .doi:10.11568/kjm.2013.21.4.365 fatcat:u75xiqlrqbb27ebizbig56x6fa