TITLE: Understanding an integral from page 15 of Titchmarsh's book "The theory of the Riemann Zeta function"
QUESTION [7 upvotes]: In Titchmarsh's book "The theory of the Riemann Zeta function" pg. 15
where the functional equation of the zeta function is being derived, 
I couldn't understand this part:
$$\frac{s}{\pi} \sum_{n=1}^{\infty} \frac{(2n\pi)^s}{n} \int_{0}^{\infty} \frac{\sin y}{y^{s+1}} dy = \frac{s}{\pi} (2\pi)^s \{-\Gamma(-s)\}\sin\frac{1}{2}s\pi\zeta(1-s)$$
I could not digest Titchmarsh's reasoning.
Can anyone explain this please?
Thanks,

REPLY [3 votes]: I do not intend to post it, but for those who are interested the proof of the tough part of this question, namely the classic result that
$$ \Gamma(s) \sin \left( \frac{\pi s}{2} \right) = \int_0^\infty y^{s-1} \sin y \textrm{ d}y $$
where $-1 < Re(s) < 1 $
can be found in "Topics in Analytic Number Theory," by Hans Rademacher (in chapter 6).