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We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

It is known that there are only finitely many normal CM-fields with class number one or with given class number (see [9, Theorem 2; 11, Theorem 2]) and J. Hoffstein showed that the degree of any normal CMfield with class number one is less than 436 (see [2, Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM-fields with class number one: there are 172 non-isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non-abelian but normal octic CM-fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM-fields, then they focussed on dihedral octic CM-fields and proved that there are 17 dihedral octic CM-fields with class number one. The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM-field with class number 2, namely: Q(V