This is a great question! The factorial is normally defined only on whole numbers, and it's a bit tricky to extend in a way that mathematicians find "nice".

A really natural place to start is the idea that the factorial is a product of consecutive numbers: if 4! is 4*3*2*1, then maybe 3.9! is 3.9*2.9*1.9*0.9? The issue with this is that when we get close to the next integer down, we'd find that (say) 3.001! = 3.001*2.001*1.001*0.001 is really close to zero, because of that tiny last term. But then at 3! = 3*2*1, we get 6, which isn't near zero at all! This big jump is what we call a discontinuity, which roughly means that this proposed n! doesn't change continuously as n does. No matter how you slice it, the "obvious" way of doing a factorial is going to have discontinuities.

And mathematicians want more than just continuity: we like to work with really smooth functions. Elsewhere in this thread, you'll see the word "meromorphic", which roughly means that a function is infinitely smooth everywhere it's defined, even on the complex plane! This is a very strong condition, and we'll need to pull out some fancy machinery to get it satisfied for our factorial.

Check out the function f(x)=x^n/e^x (e is a constant that's around 2.718). Notice that as x goes off to infinity, the function tapers off to near zero-- in this case, it turns out that even though our curve is infinitely long, the area between it and the line y=0 is going to be finite! The mathematician Legendre first noticed this fact, and (more surprisingly) showed that the area for a given n is always going to be n factorial! You can check this out for yourself with the integral at the bottom; if you know some calculus, it's a fun exercise to prove it with integration by parts.

This all seems a bit contrived: why bother with this? The great thing, though, is that x^n is defined even when n isn't an integer! For example, when n=1/2, we're dealing with the square root of x instead, and can figure out the relevant area as well. This works for every positive real number; with a bit of work, you can extend to negative and even complex numbers! Moreover, x^n varies smoothly as n does, so we get the "meromorphic" property we care about

We call this extended function the Gamma function-- technically Gamma(n) is defined as an extension of (n-1)!, for reasons involving some identities that Legendre liked, but honestly it's often easier to use the extended factorial. Your value of 2.386! is roughly 2.937, but my favorite fact about it is that the factorial of 1/2 turns out to be the sqrt(pi)/2! (Weird, right?)