I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is usually an infinite entity). I also have logarithm defined, at least in the sense of finding the regular part of it.

So, I defined the "modulus" of a divergent integral/series by analogy with complex numbers:

$$|w|=\exp(\Re(\operatorname{reg}\log w))$$

Here we take logarithm, find its regular part, find its real part and exponentiate.

To my surprise, the modulus of some of the special elements of this space turned out to be $e^{-\gamma}$ and $\frac{e^{-\gamma}}4$ (you can see the table here, modulus is in the last column).

So, I wonder, are there any other known algebraic systems, where the modulus, absolute value or norm of some special elements is fundamentally expressed via rational factors of $e^{-\gamma}$, $e^{\gamma}$ or $\pi$ (given $e^{-\gamma}$ has some roles similar to $\pi/4)$?