Inverse Laplace transform with fractional exponents

The inverse Laplace transform of

1/(1+s)^a 

with a > 0, is

e^{-t} t^{-1 + a}/Gamma(a).

I'm still trying to understand what this means, but meanwhile here's a picture. Note that it tends to an impulse when a goes to zero. When a = 1, we see the familiar exponential decay. For larger values, the inverse transform starts at 0 and reaches a maximum before decaying exponentially.

(I don't know how to make sense of the unstable case 1/(1-s)^a, so perhaps all these calculations don't help understand anything.)