# Epstein Zeta function

1. Nov 1, 2005

### eljose

Hello, i would like to know the definiton 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.

2. Nov 1, 2005

3. Nov 1, 2005

### 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.

Last edited: Nov 1, 2005