A Functional Equation

You might know the gamma function, which is one way to extend the factorial function to the complex plane. It obeys the functional equation

Γ(1) = 1

Γ(z + 1) = z Γ(z).

In fact, if we demand that it be logarithmically convex and obey the rules above, then the ordinary gamma function is in fact the only way to extend the factorial function to the positive reals. (Here Γ(n) = (n-1)!.)

Now the successor function z → z + 1 is on the lowest level of the Grzegorczyk hierarchy. The next higher level includes the function z → 2z. So, in analogy with gamma function, suppose we demand that

f(1) = 1

f(2z) = z f(z).

Then what's a "reasonable" function that satisfies this functional equation? My answer in the comments tomorrow, unless someone comes up with the same answer I did.