An infinite family of zeta functions indexed by Hermite polynomials (Q1121317)

The author considers an infinite family of zeta functions, denoted \(\zeta_{2q,\ell}(s)\), \(q, \ell =0,1,2,\dots\), defined for \(\text{Re}\, s>1+\ell\) as follows \[ \zeta_{2q,\ell}(s)=(2q)!\zeta (s-\ell)(H_q(s)-(-1)^q/q!), \] where \[ H_q(s)=\sum_{k=0}^{q-1}\frac{(-1)^k 2^{3q- 3k}(s/2)((s/2)+1) \dots ((s/2)+q-1-k)}{k! (2q-2k)!}, \] \(\zeta(s)\) being the Riemann zeta-function. Following one of the usual approaches the author proves that the function \(\zeta_{2q,0}\), \(q\geq 0\), has an analytic continuation to \(\mathbb C\setminus \{1\}\), and it satisfies the functional equation \[ \pi^{-s/2} \Gamma (s/2) \zeta_{2q,0}(s)=(-1)^q \pi^{-(1-s)/2} \Gamma ((1-s)/2) \zeta_{2q,0}(1-s). \] Moreover, \(\zeta_{2q,0}(0)=(-1)^{q+1}(2q)!/(2q!)\), and for \(m\geq 1\), \(\zeta_{2q,0}(-2m)=0\). (In the proof no knowledge of \(\zeta(s)\) is assumed). At the end of the paper the author puts forward the conjecture that for all \(q\geq 1\), the roots of \(P_q(s)=H_q(s)-(-1)^q/q!\) have real part equal to one-half, are simple, and are distinct. It is shown that this is the case for \(q=5\).