The factor $\frac12$ in the Riemann $\xi$ function:

$$\xi(s)=\frac12 s(s-1)\,\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$

was introduced by Riemann, however appears to be redundant. Once he had arrived at:

$$\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$

the remaining poles at $s=0,1$ could have been removed by the simpler factor $s\,(s-1)$. This would have also made the function entire and retains the reflection formula $\xi(s) = \xi(1-s)$.

A quite plausible explanation as to why the additional factor $\frac12$ was introduced, I found here. It boils down to the following trick:

$$\frac12s(s-1)\,\Gamma\left(\frac{s}{2}\right)=2\,\Gamma\left(\frac{s+4}{2}\right)-3\,\Gamma\left(\frac{s+2}{2}\right)$$

that serves as the first step towards deriving the well known Fourier integral expression for $\xi(\frac12+it)$. Splitting the LHS into $\Gamma$'s with constant weights, only works when the factor $\frac12$ is introduced and apparently this was a trick known to Riemann.

However, this is not the approach towards the Fourier integral that is attributed to Riemann in the well-known books about the Zeta-function of Titchmarsh (1986-edition p254/255) and Edwards (1974-edition p16/17, p41). These predominantly apply integration by parts to an integral representation of $\xi(s)$ and use a special relation involving $\psi(x)$ and $\psi'(x)$. But this approach doesn't seem to require the factor $\frac12$.

The only reference about the factor $\frac12$ that I have ever come across was in a footnote in a book or paper that (paraphrased) said: '*the factor $\frac12$ was introduced by Riemann in his 1859 paper and has since then stuck*' (unfortunately I can't recall the precise source of this quote...).

Q1: Is there any reference to literature about why the factor $\frac12$ was introduced?

Q2: Does it actually matter that the factor $\frac12$ got 'stuck' in $\xi(s)$? It is obviously just a factor, but I could imagine that such a redundant factor would make certain formula less 'beautiful', e.g without the factor, $\xi(0=\xi(1)=1$ and the Hadamard product would simply be: $\displaystyle \xi(s)=\prod_{\rho} \left(1-\frac{s}{\rho}\right)$.