While investigating the relationship
$$\Gamma\left[f,s\right]\zeta(s)=\Gamma\left[\hat{f},1-s\right] \zeta(1-s)\tag{1}$$
where $\Gamma\left[f,s\right]$ denotes the Mellin transform
$$\Gamma\left[f,s\right]=\int\limits_{-\infty}^\infty f(|x|)\, |x|^{s-1}\, dx=2 \int\limits_{-\infty}^\infty f(x)\, x^{s-1}\, dx\tag{2}$$
and $\hat{f}$ denotes the Fourier transform
$$\hat{f}(w)=\mathcal{F}_x[f(|x|)](w)=\int\limits_{-\infty }^{\infty} f(|x|)\, e^{-i 2 \pi w x} \, dx\tag{3}$$
for the functions
$$f_{1,r}(x)=2 \sqrt{r x}\, K_1\left(2\, \sqrt{r x}\right)\tag{4}$$
and
$$f_{2,r}(x)=-2 r x\, K_2\left(2\, \sqrt{r x}\right)\tag{5}$$
where $K_1(x)$ and $K_2(x)$ are modified bessel functions of the second kind, I discovered the analytic continuations
$$\frac{_3F_2\left(\frac{s}{2}+1,\frac{s}{2}+1,\frac{s}{2}+\frac{3}{2};\frac{3}{2},\frac{s}{2}+2;1\right)}{s+2}+\frac{_3F_2\left(\frac{s}{2}+\frac{1}{2},\frac{s}{2}+\frac{1}{2},\frac{s}{2}+1;\frac{1}{2},\frac{s}{2}+\frac{3}{2};1\right)}{(s+1)^2}=-\frac{\pi \csc (\pi s)}{s+1},\ \Re(s)<0\tag{6}$$
and
$$_3F_2\left(\frac{1}{2},\frac{s}{2}+\frac{3}{2},\frac{s}{2}+2;\frac{3}{2},\frac{3}{2};1\right)+\frac{_3F_2\left(\frac{s}{2}+\frac{3}{2},\frac{s}{2}+\frac{3}{2},\frac{s}{2}+2;\frac{3}{2},\frac{s}{2}+\frac{5}{2};1\right)}{s+3}=\frac{\pi \tan \left(\frac{\pi s}{2}\right)}{2 s+4},\ \Re(s)<-1\tag{7}$$
which are illustrated in Figures (1) and (2) below.
Question (1): Are the two analytic continuations defined in formulas (6) and (7) above two specific examples of a more general class of analytic continuations and if so, is there a formula for this generalization?
Question (2): Are there any other known analytic continuations related to the Hypergeometric $_3F_2$ function?
Figure (1) below illustrates the left-side of formula (6) in orange overlaid on the right-side of formula (6) in blue.
Figure (1): Illustration of left and right sides of formula (6) in orange and blue
Figure (2) below illustrates the left-side of formula (7) in orange overlaid on the right-side of formula (7) in blue.
Figure (2): Illustration of left and right sides of formula (7) in orange and blue
The relationship illustrated in formula (1) above is valid for any Schwartz function $f$, but it always simplifies to the Riemann zeta functional equation
$$\zeta(s)=\frac{\Gamma(\hat{f},1-s)}{\Gamma(f,s)}\,\zeta(1-s)=2^s\, \pi^{s-1}\, \sin\left(\frac{\pi s}{2}\right)\, \Gamma(1-s)\,\zeta(1-s)\tag{8}.$$
It also works with (at least some) distributions and non-Schwartz functions where perhaps the optimal choice is $f(x)=\delta(x-1)$ in which case
$$\hat{f}(w)=\mathcal{F}_x[f(|x|)](w)=2 \cos (2 \pi w)\tag{9}$$
and $\Gamma(f,s)=2$.