On the Wiki-page about the Dirichlet $\beta$-function, it is noted that "this function is also the simplest example of a series non-directly related to $\zeta(s)$ which can also be factorized as an Euler product."
The $\beta$-function is defined as:
$$\beta(s)=\displaystyle \small \sum_{n\ge0}\frac{(-1)^n}{(2n+1)^s}=\frac{1}{\Gamma(s)}\int_0^\infty x^{s-1}\frac{e^{-x}}{e^{-2x}+1}dx= 4^{-s}\big(\zeta\left(s,\frac14\right)-\zeta\left(s,\frac34\right)\big) $$
hence it is directly related to the Hurwitz zeta function (but not to $\zeta(s)\,$). However, after applying Parseval's theorem to the Fourier/Laplace transform of the integral for $\Gamma(s)\,\beta(s)$ I get:
$$\frac{1}{\pi}\int_{0}^{\infty} \beta\left(s+x\,i\right)\Gamma\left(s+x\,i\right)\beta\left(s-x\,i\right)\Gamma\left(s-x\,i\right)dx=\int_0^\infty x^{2s-1}\frac{e^{2x}}{(e^{2x}+1)^2}dx$$
for $s \in \mathbb{C}, \Re(s)>0$. The RHS can be simplified into the closed form: $$\frac{\Gamma(2s)}{2^{2s}}\,\eta(2s-1)$$ with $\eta(z)=(1-2^{1-z})\,\zeta(z)$ the Dirichlet $\eta$-function.
So, it seems that $\Gamma(s)\,\beta(s)$ could be transformed into an expression with $\Gamma(2s)$ and $\zeta(2s-1)$ and I wondered whether a reverse route exists as well (i.e. to transform $\zeta(s)$ back into $\beta(s)\,$ to reflect maybe duality)? Does e.g. a "square root" equivalent for the Parseval theorem exist?