A question from Titchmarsh's zeta function book. On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that:
$\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get that it should vanish, shouldn't it?
What's wrong here?
https://books.google.co.il/books?id=1CyfApMt8JYC&printsec=frontcover#v=onepage&q&f=false

For the functional equation 
$$
    \xi(s) = \xi(1 - s) 
$$
we have the definition
$$
    \xi(s) = \frac{1}{2}\pi^{-s/2}s(s-1)\Gamma\left(\frac{s}{2}\right)\zeta(s).\! 
$$
The functional equation just gives $\xi(0)=\xi(1)$. The function $Z(s)=\frac{1}{2}\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$ has a meromorphic continuation to the whole $s$-plane, with simple poles at $s=0$ and $s=1$.
So the question is, what the value of $\xi(0)$ is. Following your argument, we should have $\xi(0)=0$, and not $\xi(0)=1/2$. However, we have to take into account the simple poles, too. 
