Tetration series for integer exponent. Can you find the pattern?

[marraco]

got stuck with the question how to define the half-derivative of the zeta at zero (based on the Dirichlet-series representation) and this question is still open in mathoverflow...

Did you tried to make a Laplace transform? (if that exist for a matrix)

then an derivative is just a product with a simple function raised to n, and a fractional derivative is just a fractional value for n.

Ok, this may not be one standard definition of fractional derivative, but maybe it leads somewhere.