This question assumes the definition of the modular discriminant $\Delta(\tau)$ in (1) below with Fourier series representation defined in (2) below where $\tau(n)$ is the Ramanujan tau function.
(1) $\quad\Delta(\tau)=q\prod\limits_{r=1}^\infty(1-q^r)^{24}\,,\quad q=e^{2 \pi i \tau}$
(2) $\quad\Delta(\tau)=\sum\limits_{n=1}^\infty\tau(n)\,e^{2 \pi i n \tau}$
I believe $\Delta(i \sigma)$ can be evaluated via formula (2) below where $b(n)$ is the Dirichlet convolution of $\tau(n)$ and $\mu(n)$, i.e. $b(n)=\tau(n)*\mu(n)=\sum\limits_{d|n}\tau(d)\,\mu\left(\frac{n}{d}\right)$.
(3) $\quad\Delta(i \sigma)=\sum\limits_{n=1}^\infty b(n)\,(\coth(\pi n \sigma)-1)\,,\quad\sigma>0$
The following figure illustrates formula (3) above for $\Delta(i \sigma)$ in orange overlaid on the reference function $\Delta(i \sigma)=\eta(i \sigma)^{24}$ in blue where the series in formula (3) above is evaluated over the first $200$ terms.
Figure (1): Illustration of formula (3) for $\Delta(i \sigma)$
More generally if $f(\tau)$ is a modular form with Fourier series illustrated in (4) below, I believe $f(i \sigma)$ can be evaluated via formula (5) below where $b(n)$ is the Dirichlet convolution of $a(n)$ and $\mu(n)$, i.e. $b(n)=a(n)*\mu(n)=\sum\limits_{d|n}a(d)\,\mu\left(\frac{n}{d}\right)$.
(4) $\quad f(\tau)=a(0)+\sum\limits_{n=1}^\infty a(n)\,e^{2 \pi i n \tau}$
(5) $\quad f(i \sigma)=a(0)+\sum\limits_{n=1}^\infty b(n)\,(\coth(\pi n \sigma)-1)\,,\quad\sigma>0$
Question: Is formula (3) for $\Delta(i \sigma)$ above (and the generalization to other modular forms in formula (5) above) a known and proven relationship and if so, can a reference be provided?
