On the value at s=-1 of partial zeta functions for ray classes of real quadratic field

Let k be a real quadratic field. Let a be an integer of k and m be a pos­itive rational integer. Denote by ζ((a), (m), s) be a partial zeta function associated to a ray class containing the principal ideal (a) which is defined with a conductor (m). We give a formula of the value of ζ((a), (m), -1) by means of the modifica— tion of the sum defined by L. Carlitz, c.f. [2]. It is an analogue to the generalized Dedekind sum defined in [6].