$\mathcal{X}\left(f\right):$ $\zeta \left(-2\pi i\alpha +\frac{1}{2}\right)={\sum }_{|x|=1}^{|x|=\infty }{|x|}^{2\pi i\alpha -\frac{1}{2}}$ $\stackrel{{U}_{x}^{\xi ,r}:}{\overbrace{\to }}$ ${U}_{f}^{\xi ,r}\mathcal{X}\left(f\right)={a}^{r+1}{\text{e}}^{-2\pi i\xi \left(1-a\right)f}\mathcal{X}\left(af\right):$ ${\sum }_{|x|=1}^{|x|=\infty }{\text{e}}^{-2\pi i\alpha |x|}$ $↓$ (warping operator, $r=-\frac{1}{2}$ , $\xi =0$ , $a={\text{e}}^{-x}$ ; ${U}_{x}^{\xi =0,r=-\frac{1}{2}}={U}_{x}$ ) $↓$ ${\mathcal{M}}_{f}^{\xi }\left[\mathcal{X}\left(f\right)\right]=\mathcal{F}{U}_{n}^{\xi ,r}\left[\mathcal{X}\left(f\right)\right]:$ $\mathcal{F}{\sum }_{|x|=1}^{|x|=\infty }{\text{e}}^{-2\pi i\alpha |x|}={\sum }_{|x|=1}^{|x|=\infty }\delta \left(|x|-\alpha \right)$ with $\mathcal{F}$ the Fourier Transform with ordinary unitary frequency $\underset{{\mathcal{M}}_{x}}{\underbrace{\to }}$ $\begin{array}{l}{\mathcal{M}}_{f}^{\xi }\left[{U}_{x}^{\xi ,r}\mathcal{X}\right]\left(\beta \right)\\ ={a}^{-2\pi i\beta }{\mathcal{M}}_{f}^{\xi }\left[\mathcal{X}\right]\left(\mu \right)\cdot {\int }_{0}^{\infty }{\sum }_{|x|=1}^{|x|=\infty }{\text{e}}^{-2\pi i\beta |x|}{|x|}^{\mu -1}\text{d}x\\ ={\int }_{0}^{\infty }{\sum }_{|x|=1}^{|x|=\infty }\delta \left(|x|-\beta \right){|x|}^{\mu -1}\text{d}x\\ =\zeta \left(1-\mu \right)\end{array}$ The result is equivalent with $-{1}^{-\mu }{i}^{\mu }\frac{\Gamma \left(\mu +\alpha \right)}{\Gamma \left(\alpha \right)}\varsigma \left(\mu +\alpha ,\beta \right)$ for $\xi \ne 0$ , $\mu =2\pi i\beta +r$ , $a={\text{e}}^{-x}$ and initial input $\varsigma \left(\alpha \right)$ . Hence the factor ${a}^{-2\pi i\beta }$ may be moved to the integrand of the found Fourier Transform $\begin{array}{l}{\int }_{-\infty }^{\infty }{\sum }_{|f|=1}^{|f|=\infty }{|f|}^{-\alpha }{\text{e}}^{-2\pi i\left(\xi +\beta \right)f}{f}^{\mu }\text{d}f\\ =2\pi {i}^{-\mu }{\sum }_{|\xi |=1}^{|\xi |=\infty }\frac{\Gamma \left(\mu +\alpha \right)}{\Gamma \left(\alpha \right)}{|\xi +\beta |}^{-\mu -\alpha }\\ =2\pi {i}^{-\mu }\frac{\Gamma \left(\mu +\alpha \right)}{\Gamma \left(\alpha \right)}\left\{\varsigma \left(\mu +\alpha ,\beta \right)-{|\beta |}^{-\mu -\alpha }\right\}\end{array}$ (71), with $\varsigma \left(s,\lambda \right)$ the Hurwitz Zeta function (100).