We study the Artin L-function L(s, χ) associated to the unique character χ of degree 2 in quaternion fields of degree 8. We first explain how to find examples of quaternion octic fields with not too large a discriminant. We then develop a method using a quick compuation of the order n*#967; of the zero of L(s, χ) at the point s = 1/2. In all our calculations, we find that nχ only depends on the sign of the root number W(χ); indeed nχ = 0 when W (χ) = -1. Finally we give some estimates on nχ and low zeros of L(s, χ) on the critical line in terms fo the Artin conductor *#402;χ of the character χ.
"On Artin L-Functions for Octic Quaternion Fields." Experiment. Math. 10 (2) 237 - 246, 2001.