I was experimenting with the following products over all primes:
$$\prod_p \left(\frac{p^s}{{p^s-\sin\left(\dfrac{p^s \,\pi}{2}\right)}} \right)\,\cdot\,\prod_p \left(\frac{p^s+\sin\left(\dfrac{p^s \,\pi}{2}\right)}{p^s} \right)=\prod_p \left(\frac{p^s+\sin\left(\dfrac{p^s \,\pi}{2}\right)}{{p^s-\sin\left(\dfrac{p^s \,\pi}{2}\right)}} \right)$$
and used $p^s$ instead of $p$ in the Dirichlet character, i.e.: $\chi_{4}(p^s)=\sin\left(\dfrac{p^s \,\pi}{2}\right)$.
I then found that each factor on the LHS can be expressed in a closed form at integer values comprising of a rational and $\pi^n$. The RHS is purely rational and has a closed form as well:
For $s= $ even integer:
$$\large \lambda(s) \qquad \cdot \qquad \frac{\lambda(s)}{\lambda(2s)} \qquad=\qquad\frac{\lambda(s)^2}{\lambda(2s)} $$
$s=2 \large \qquad \qquad\qquad \qquad \frac{\pi^2}{8} \qquad \cdot \qquad \frac{12}{\pi^2} \qquad=\qquad \frac{3}{2}$ $s=4 \large \qquad \qquad\qquad \qquad \frac{\pi^4}{96} \qquad \cdot \qquad \frac{1680}{17\pi^4} \,\quad=\qquad \frac{35}{34}$ $s=6 \large \qquad \qquad \qquad \qquad \frac{\pi^6}{960} \quad \,\,\, \cdot \qquad \frac{665280}{691\pi^6} \,\,\,=\qquad \frac{693}{691}$
$\dots$
For $s= $ odd integer:
$$\large \beta(s) \qquad \cdot \qquad \frac{\beta(s)}{\lambda(2s)} \qquad=\qquad\frac{\beta(s)^2}{\lambda(2s)} $$
$s=1 \large \qquad \qquad\qquad \qquad \frac{\pi}{4} \qquad \,\,\, \cdot \qquad \frac{2}{\pi} \,\,\qquad=\qquad \frac{1}{2}$ $s=3 \large \qquad \qquad\qquad \qquad \frac{\pi^3}{32} \qquad \cdot \qquad \frac{30}{\pi^3} \qquad=\qquad \frac{15}{16}$ $s=5 \large \qquad \qquad\qquad \quad \,\,\,\,\,\frac{5\pi^5}{1536} \quad \,\, \cdot \qquad \frac{9450}{31\pi^5} \quad \,=\qquad \frac{7875}{7936}$
$\dots$
where I used the Dirichlet $\lambda$- and Dirichlet $\beta$-functions, that each have closed form expressions.
Question:
Just like $\displaystyle \prod_p \left(\frac{p^s+1}{p^s-1}\right)=\frac{\zeta(s)^2}{\zeta(2s)}$ is valid for all $s \in \mathbb{C}, \Re(s)>1$, could there also be a way to extend the closed forms above towards non-integer values with $\Re(s)>\frac12$?