I have also asked this question here. I am posting it on this forum as well in order to increase the number of people who see this question.
Consider the (euclidean) path integral for the free massive scalar field in $d$ dimensions, giving the partition function
$$Z_m=\int\mathcal{D}\phi~e^{\int dx^d~\phi(\Delta-m^2)\phi}$$
with Laplace-Beltrami operator
$$\Delta = \sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}.$$
E.g. on the torus we can expand in Fourier modes
$$\phi = \sum_{k_1,k_2,...,k_d\in\mathbb{Z}}e^{i \vec{k}\cdot \vec{x}}\psi_{\vec k}$$
where $\psi_{\vec{k}}{}^*=\psi_{-\vec{k}}$ since $\phi$ is real, and which leads to
$$\mathcal D \phi \sim \prod_{k_1,k_2,...,k_d\in\mathbb{N}}d\text{Re}(\psi_\vec{k})d\text{Im}(\psi_\vec{k})$$
and
$$\begin{align}\int dx^d~\phi(\Delta-m^2)\phi =& -\sum_{k_1,k_2,...,k_d\in\mathbb{Z}}(\vec{k}\cdot\vec{k}+m^2)|\psi_{\vec{k}}|^2\\=& -\sum_{k_1,k_2,...,k_d\in\mathbb{Z}}(\vec{k}\cdot\vec{k}+m^2)(\text{Re}(\psi_{\vec{k}})^2+\text{Im}(\psi_{\vec{k}})^2)\end{align}$$
Therefore, using Gaussian integration we get
$$\begin{align}Z_m\sim& \prod_{k_1,k_2,...,k_d\in\mathbb{N}}\sqrt{\frac{\pi}{\vec{k}\cdot\vec{k}+m^2}}\sqrt{\frac{\pi}{\vec{k}\cdot\vec{k}+m^2}}\\ =&\prod_{k_1,k_2,...,k_d\in\mathbb{N}}\frac{\pi}{\vec{k}\cdot\vec{k}+m^2}\end{align}$$
up to some numerical factor.
At this point the references I'm aware of either claim victory or throw in the towel, even though the infinite product representation of this result is just as inconvenient for numerical evaluation as the original path integral form on the first line.
I imagine this functional determinant is one of the simpler ones mathematicians have encountered. My question is:
Has this functional determinant been actually evaluated in various dimensions? What is the resulting function and the explicit $m$ dependence?