What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

Question

As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis:

$$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$

My question is, what "abstract" properties does this function have (like for example the $\exp$-function has the property $\exp(x+y) = \exp(x)\exp(y)$) and do these properties characterize this function?

Thanks for your help! (Also I am not sure how to properly tag this question)

Related: A group theoretic interpretation of Lagarias inequality

Edit: Another related question: Dou you know of any other situation where the function $L(x)$ occurs? That would also be quite interesting!

Second Edit: I found a very exciting connection to "Logarithmic numbers" as defined by J. M. Gandhi:

http://oeis.org/A002741

The numbers $\frac{d^n}{dx^n} L(x)$ at $x=1$ are related to the Logarithmic numbers as defined by Gandhi.

There are two papers by Gandhi, on this topic:

http://oeis.org/A002741/a002741.pdf which is a little bit hard to read because it is scanned, and https://www.tandfonline.com/doi/abs/10.1080/00029890.1966.11970871 where there is made a connection to $\sigma(n)$ and the logarithmic numbers.

Third edit (18.05.2019): I think I found a very interesting property which seems to always hold:

If $A$ is a normal ($A^TA=AA^T$) and non-singular matrix and such that $\frac{1}{|A|}\cdot A$ is a doubly stochastic, positive matrix, then we have:

$$L(|A|) = |L(A)|$$ and $L(A)$ is a normal, non-singular matrix such that $\frac{1}{|L(A)|}\cdot L(A)$is doubly stochastic, positive matrix. where $|.|$ denotes the spectral norm.

To be more concrete I will tell how I construct the matrix for a given finite group $G$:

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of the group $G$:

$$ a_{i,j} = \frac{1}{1+|g_i g_j^{-1}|}$$

This is a group matrix as defined by Dedekind and Frobenius. Let $H_G:= \sum_{g \in G} \frac{1}{|g|+1}$ be the harmonic number associated to $S$ and $G$.

Here are my conjectures concerning this matrix some of which I can prove:

1. $H_G = |A|$ [proved by Perron-Frobenius theorem]

2. (If 1. is true, then by definition of $A$ we must have that $1/H_G A = 1/|A| A $ is a doubly stochastic matrix [thats clear by 1. and definition of $A_G$.]

3. $A = \sum_{g \in G} \frac{1}{1+|g|} \rho(g)$ is the Birkhoff-Neunmann decomposition induced by the doubly stochastic matrix [proved by definition of $A_G$ and $\rho$]

4. Using 2. I can prove that $A$ is a normal matrix

5. $A$ is non-singular. [that remains mysterious]

My updated question is, if (any of) this can be proven (or if it is known, then any reference would also be nice)?

Thanks for your help!