The Family of the Third Kind $\{{\mathfrak 3} (s | \tau)\}$

Abstract

From Sect. 5.3, we recall the definition [51, 106]

$$\mathfrak{Z}(s | \tau) = \sum\limits_{k = 1}^{\infty}(\tau_{k} + \tau)^{-s},\qquad {\rm Re}\,\, s > 1.$$

(9.1)

For general $\tau (\neq 0), \mathfrak{Z}(s | \tau)$ is built on a one-sided set of zeros, say $(\frac{1}{2} + {\rm i}\tau_{k})$ only; hence it seems harder to tackle than the other two families, and the earlier studies (just quoted) have revealed more singular features indeed.