The Epstein Zeta Function: Definition, Zeros, and Functional Equation Explained

[eljose]

Hello, i would like to know the definition of epstein zeta function and its zeros,i would like to know if s is a zero then s* is also a zero and what would be the functional equation that the Epstein function satisfy...thanks.

 

[Zurtex]

http://mathworld.wolfram.com/EpsteinZetaFunction.html

http://www.pubmedcentral.nih.gov/pagerender.fcgi?artid=1063041&pageindex=1

http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.nmj/1118798307

http://www.iop.org/EJ/abstract/0264-9381/8/1/011

http://www.google.co.uk/search?hs=a...ial&q=Epstein+Zeta+Function&btnG=Search&meta=

 

[eljose]

Thanks..for the links but i have not been able to see a link that gives you the functional equation for Epstein series...

EDIT: i have finally found the functional equation for Epstein Zeta function $$Zeta_{Q}(s)$$ in the form:

$$\pi^{-s}\Gamma(s)Z_{Q^{-1}}(s)=|Q|^{1/2}\pi^{s-n/2}\Gamma(n/2-s)Z_{Q}(n/2-s)$$ (1)

with |Q|=Det(Q), where for n=2 Q is the matrix of the quadratic form Q(x,y)=ax^2+by^2+cxy, with n=2 and Q=Q^{-1} then |Q|=1 for the cases n=2 and Q=Q^{-1}=R the functional equation (1) is exactly equal to Riemann functional equation for the function $$\zeta(s)$$ my question is if RH would hold also for this case of the Epstein function as for the case of the Riemann zeta function.