In the shaded section below, I listed 8 well-known functions that nicely hang together:
$$\displaystyle\frac{\beta(s)+\lambda(s)}{\kappa(s)+\alpha(s)}=\frac{\zeta_H\left(s,\frac12\right)+\eta_H\left(s,\frac12\right)}{\zeta(s)+\eta(s)}$$
Each of these functions has a Laplace/Fourier integral transform that can be "plugged into" the integral from the Parseval theorem:
$$\small G_f(s):=\frac{1}{\pi\,\Gamma(s+1)}\int_{0}^{\infty} f\left(\frac{s+1}{2}+x\,i\right)\Gamma\left(\frac{s+1}{2}+x\,i\right)f\left(\frac{s+1}{2}-x\,i\right)\Gamma\left(\frac{s+1}{2}-x\,i\right)dx$$
Based on the example for $f(s)=\zeta(s)$ shown here, I managed to derive the new closed forms for each of the 8 functions and also listed them in the shaded area (note: only analytically continued to $\Re(s)>0$). All closed forms $G_f(s)$ are now directly related to $\zeta(s)$ (note that f.i. $\beta(s)$ wasn't).
I then found that the same nice relation remains for $s \in \mathbb{C}$ and $\Re(s)>0$, i.e:
$$\displaystyle\frac{G{_\beta}(s)+G_{\lambda}(s)}{G_{\kappa}(s)+G_{\alpha}(s)}=\frac{G_{\zeta_H}\left(s,\frac12\right)+G{\eta_H}\left(s,\frac12\right)}{G_{\zeta}(s)+G_{\eta}(s)}$$
1) Is the fact that this relation remains the same a logical consequence of applying the Parseval theorem? (note that other relations like $\zeta(s)+\eta(s)=2\lambda(s)$ or $\alpha(s)+\lambda(s)=\zeta(s)$ do break down).
2) Besides $\zeta(s)$ and sums of $\zeta_H(s)$, the $\lambda$ and the $\beta$-functions are as $L(s,\chi_{4,1})$ and $L(s,\chi_{4,2})$ respectively, the only ones related to a Dirichlet L-series. However, I could not find any further connections like these in the "G-world". Do closed forms for L-series (other than mod 4) also exist after applying the Parseval theorem?
Addition to question 2:
I believe that I have now also found a closed "G"-form for the Dirichlet L-series.
With $\chi_{_{k,\,j}}(n)$ being a Dirichlet character with modulus $k$ and index $j$, it is well known that:
$$\displaystyle L(s,\,\chi_{_{k,\,j}})=\frac{1}{k^s}\sum_{n=1}^k \chi_{_{k,\,j}}(n)\cdot\zeta_H \left(s,\frac{n}{k}\right)$$
Then I defined:
$$\zeta_H^*(s,a)= \zeta_H(s,2a)-(2a-1)\,\zeta_H(s+1,2a)$$ and found that:
$$\displaystyle G_{L(s,\,\chi_{{k,\,j}})}=\frac{1}{k^{s+1}}\left(\sum_{n=1}^{k} \chi_{_{k,\,j}}(n)^2\cdot\zeta_H^* \left(s,\frac{n}{k}\right)+2\sum_{n=1}^{k-2}\sum_{m=n+1}^{k-1}\chi_{_{k,\,j}}(n)\cdot\chi_{_{k,\,j}}(m)\cdot\zeta_H^* \left(s,\frac{n+m}{2k}\right)\right)$$
that seems valid for $\Re(s)>-1$, except for index $j=1$, when it is valid for $\Re(s)>1$ only. The latter can be analytically continued to f.i. $0 < \Re(s) <1$ by just subtracting from the above:
$$\displaystyle \small \frac{2}{s\,k}\,\left(L(s,\,\chi_{{k,1}})+\frac{1}{k^s}\sum_{n=1}^{k-2}\sum_{m=1}^{k-n-1}\chi_{_{k,1}}(m)\cdot\chi_{_{k,1}}(m+n)\cdot\left(\zeta_H \left(s,\frac{m}{k}\right)+\zeta_H \left(s,\frac{m+n}{k}\right)\right)\right)$$
$\\$ $\\$
$\displaystyle \small \sum_{n\ge1}\frac{1}{n^s}=$ Riemann $\zeta(s)$-function
$\qquad \qquad \rightarrow \small G_\zeta(s)= \begin{cases} \zeta(s) - \zeta(s+1) & \qquad \qquad \qquad \qquad \Re(s)>1 \\ \\ \zeta(s)-\zeta(s+1)-2\,\frac{\zeta(s)}{s}& \qquad \qquad \qquad \qquad 0<\Re(s)<1 \end{cases}\\$
$\displaystyle \small \sum_{n\ge1}\frac{1}{(2n)^s}=$ Self named $\alpha(s)$-function $\small =2^{-s}\zeta(s)$ $\qquad \qquad \rightarrow \small G_\alpha(s)= \begin{cases} \frac{1}{2^{s+1}}\big(\zeta(s)-\zeta(s+1)\big) & \qquad \qquad \quad \quad \,\, \Re(s)>1 \\ \\ \frac{1}{2^{s+1}}\big(\zeta(s)-\zeta(s+1)\big)-\frac{\alpha(s)}{s} & \qquad \qquad \quad \quad \,\, 0<\Re(s)<1 \end{cases}\\$
$\displaystyle \small \sum_{n\ge0}\frac{1}{(2n+1)^s}=$ Dirichlet $\lambda(s)$-function $=\small (1-2^{-s})\zeta(s)$
$\qquad \qquad \rightarrow \small G_\lambda(s)= \begin{cases} \frac{1}{2^{s+1}}\zeta(s)& \qquad \qquad \quad \qquad \qquad \qquad \,\,\, \Re(s)>1 \\ \\ \frac{1}{2^{s+1}}\zeta(s)-\frac{\lambda(s)}{s} & \qquad \qquad \qquad \quad \qquad \qquad \,\,\, 0<\Re(s)<1 \end{cases}\\$
$\displaystyle \small \sum_{n\ge 0}\frac{1}{(a+n)^s}=$ Hurwitz $\zeta_H(s,a)$-function $=\small \zeta_H\left(s,\frac12\right) = (1-2^s)\zeta(s)$
$\qquad \qquad \rightarrow \small G_{\zeta_H}(s)= \begin{cases} \zeta_H(s,2a)-(2a-1)\,\zeta_H(s+1,2a) & \,\,\,\, \Re(s)>1 \\ \\ \zeta_H(s,2a)-(2a-1)\,\zeta_H(s+1,2a)-2\,\frac{\zeta_H(s,a)}{s} & \,\,\,\, 0<\Re(s)<1 \end{cases}\\$
$\displaystyle \small \sum_{n\ge1}\frac{(-1)^{n-1}}{n^s}=$ Dirichlet $\eta(s)$-function $\small=(1-2^{1-s})\zeta(s)$
$\qquad \qquad \rightarrow \small G_\eta(s) = \eta(s+1)-\eta(s) \,\, \qquad \qquad \qquad \qquad \qquad \qquad \,\,\,\,\Re(s)>0\\$
$\displaystyle \small \sum_{n\ge1}\frac{(-1)^{n-1}}{(2n)^s}=$ Self named $\kappa(s)$-function $=\small 2^{-s}\eta(s)$
$\qquad \qquad \rightarrow \small G_\kappa(s)=\frac{1}{2^{s+1}}\big(\eta(s+1)-\eta(s)\big) \qquad \qquad \qquad \quad \qquad \quad \,\,\, \Re(s)>0\\$
$\displaystyle \small \sum_{n\ge0}\frac{(-1)^n}{(2n+1)^s}=$ Dirichlet $\beta(s)$-function $\small =4^{-s}\big(\zeta\left(s,\frac14\right)-\zeta\left(s,\frac34\right)\big)$
$\qquad \qquad \rightarrow \small G_\beta(s) = \frac{1}{2^{s+1}}\,\eta(s) \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \, \Re(s)>0\\$
$\displaystyle \small \sum_{n\ge 0}\frac{(-1)^n}{(a+n)^s}=$ Self named $\eta_H(s,a)$-function $=\small 2^{-s}\big( \zeta_{H}\left(s,\frac{a}{2}\right)-\zeta_{H}\left(s,\frac{a+1}{2}\right)\big)$
$\qquad \qquad \rightarrow \small G_{\eta_H}(s) = \eta_H(s,2a)-(2a-1)\,\eta_H(s+1,2a) \,\, \qquad \qquad \quad \,\Re(s)>0\\$
