What is the smallest factorial non-integer?

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For those who don't know factorial system, it works like this:
! means "factorial"
one factorial is 1!=1
2!=2•1=2
3!=3•2•1=6
.
.
.
10!=10•9•8•7•6•5•4•3•2•1=3,628,800
etc etc

Factorials of negative numbers are undefined, and so are factorials of complex numbers (#i+real numbers).

Apparently, factorials work for positive NON-integers. I didn't know until I tried typing it into a calculator.
Now, 0!=1 and 1!=1 , but x! is less than 1 if x is less than 1 .

The factorials of non-integers greater than 1 is as follows.
3.4! = 3.4 • 2.4 • 1.4 • (0.4!)
1.7! = 1.7 • (0.7!)

But how does the calculator solve for factorials less than one? ( 0.4! and 0.7! in the examples.)

The lowest factorial I can find is 0.436594! = 0.87617767556448
What is the significance of these two numbers and why are they the lowest?
(The resolution of my calculator prevents me from investigating further digits. 0.436595! gives me the same answer.)
Is the actual lowest factorial number rational or irrational?

This web calculator may help if you want to try it yourself –> http://web2.0calc.com

Sort of. Factorial itself is not defined for anything but positive integers, but there's a gamma function:

https://en.wikipedia.org/wiki/Gamma_function

Interesting. So if I graph the derivative of the Gamma function, I wonder if it will cross the x-axis at 1 + 0.436594.
I may need to investigate with a graphing calculator when I'm less busy.

Ah wow somone got ehre with the gamma function first.

Oh, there's Gamma min right on the wikipedia page. (Thanks again for directing me to the link.)
I can't embed the images for the graphics, but here are the urls anyway

https://upload.wikimedia.org/math/1/e/a ... 0a52be.png
https://upload.wikimedia.org/math/2/0/3 ... cc5322.png

So there is some verification that I was on the right track with those numbers for Gamma min,
but I'm still not sure I understand why they arise there,
instead of say at (1/e)! , Gamma(1+1/e)
or (0.5^0.5)! , Gamma(1+0.5^0.5)
or (golden ratio - 1)! , Gamma(golden ratio)
or some other number that occurs elsewhere in nature.
Perhaps this number for Gamma min occurs elsewhere in nature
like those other irrational numbers listed. If so I am curious to know where.

(By recurring, I mean by virtue of it's mathematical properties,
not because we look for it until we find it by coincidence.)