We can simplify this problem by applying a transformation, because the values on your die are a simple linear function of the values on a standard die labeled $1$ to $4$. That is, if you take a die labeled $[1,2,3,4]$, and replace each face labeled $x$ with $0.4x+2.3$, then the resulting labels are $[2.7,3.1,3.5,3.9]$. It should be obvious that six copies of a standard die can roll any number between $6$ and $24$ inclusive. This means that six copies of your die can roll any number of the form $0.4 x+6\times 2.3$, where $x$ can be any integer between $6$ and $24$, inclusive.
This implies that for any sum, $S$, the number of ways that six of your dice can sum to $S$ is equal to the number of ways that a standard die with labels $[1,2,3,4]$ can sum to
$$
2.5(S - 6\times 2.3)
$$
All that remains is, how do you compute the number of ways to roll each possible sum with six standard four-sided dice? This can indeed be done in Excel.
Enter the number $1$ into cells $A5,A6,A7,A8$.
Enter the formula =SUM(A1:A4) into cell $B5$.
Click and drag to fill the formula in $B5$ to all of the cells between $B5$ and $F28$.
The column $F$ will now contain the number of ways to roll each possible sum with six standard dice. Specifically, row number $r$ has the number of ways to roll $r-4$. If you then divide each entry in column $F$ by $4^6$, you get the probability of attaining each roll.
