Mean values of derivatives of L-functions in function fields: I

We investigate the first moment of the second derivative of quadratic Dirichlet L-functions over the rational function field. We establish an asymptotic formula when the cardinality of the finite field is fixed and the genus of the hyperelliptic curves associated to a family of Dirichlet L  -functions over Fq(T)Fq(T) tends to infinity. As a more general result, we compute the full degree three polynomial in the asymptotic expansion of the first moment of the second derivative of this particular family of L-functions.