analysis - Riemann's thinking on symmetrizing the zeta functional equation

In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as

$$\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\tfrac12\pi s)\zeta(1-s).$$

and says three lines later that it may be expressed as

$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)=\xi(1-s).$$

After reading MO-Q7656, I started wondering whether, as his ideas evolved before he wrote the paper, he first constructed $\xi(s)$ by noticing that multiplying $\zeta(s)$ by $\Gamma(\frac{s}{2})$ introduces a simple pole at $s=0$ thereby reflecting the pole of $\zeta(s)$ at $s=1$ through the line $s=1/2$ and that the other simple poles of $\Gamma(\frac{s}{2})$ are removed by the zeros on the real line of the zeta function. The $\pi^{-s/2}$ can easily be determined as a normalization by an entire function $c^s$ where $c$ is a constant, using the complex conjugate symmetry of the gamma and zeta fct. about the real axis.

Anyone familiar with how his ideas (thinking) evolved?

(Update 5/13/2012) Riemann in his fourth equality in his paper, before he writes down the functional eqn., has

$$2sin(\pi s)(s-1)!\zeta(s)=i\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x-1}dx$$

where the contour sandwiches the positive real axis and surrounds the origin in the positive sense.

For $m=0,1,2, ...,$ this gives

$$\zeta(-m)=\frac{(-1)^{m}}{2\pi i}\oint_{|z|=1}\frac{m!}{z^{m+1}}\frac{1}{e^z-1}dz=\frac{(-1)^{m}}{m+1}\frac{1}{2\pi i}\oint_{|z|=1}\frac{(m+1)!}{z^{m+2}}\frac{z}{e^z-1}dz$$

from whence you can see, if you are familiar with the exponential generating fct. for the Bernoulli numbers, that the integral vanishes for even $m$. Riemann certainly was familiar with these numbers and states that the integral relation he gives implies the vanishing of $\zeta(s)$ for $m$ even (but gives no explicit proof). Edwards in Riemann's Zeta Function (pg. 12, Dover ed.) even speculates that ".. it may well have been this problem of deriving (2) [Euler's formula for $\zeta(2n)$ for positive $n$] anew which led Riemann to the discovery of the functional equation ...."

Riemann gives two proofs of the fct. eqn.--the first based on contour integration and the singularities of $\frac{1}{e^z-1}$, the second, on the theta function. Edwards wonders:

"Since the second proof renders the first proof wholly unnecessary, one may ask why Riemann included the first proof at all. Perhaps the first proof shows the argument by which he originally discovered the functional equation or perhaps it exhibits some properties which were important in his understanding of it."

In fact, Riemann states on page 3 of his paper, "This property of the function [$\xi(s)=\xi(1-s)$] induced me to introduce, in place of $(s-1)!$, the integral $(s/2-1)!$ into the general term of the series" for zeta. And then he proceeds to introduce the Jacobi theta function.

Edit Dec 18, 2014

In the early 1820s, both Abel and Plana separately published what is now called the Abel-Plana formula (see Wikipedia). In the title of Plana's, he mentions the Bernoulliens. I wonder how much Riemann was influenced by these papers.