I tried summing the first 100 of them and initially looks like the sum is blowing up. However that could be due to some numerical problem with Mathematica computing the high-ordered derivatives correctly. I computed this sum:

$\displaystyle \sum_{n=0}^{25}\left( \frac{d^n \zeta(z)}{dz^n}\biggr|_{z=i \gamma_1}\right)$

where $\displaystyle \gamma_1$ is the imaginary part of the first non-trivial zero. I used the following Mathematica code and plotted the real part of partial sums below.

Code:

mynum = Im[ZetaZero[1]];
mytable = Table[
N[Re[Sum[D[Zeta[z], {z, n}],
{n, 0, nmax}] /. z -> I*mynum]],
{nmax, 1, 25}]

That doesn't mean it's not zero of course. Only that the first numerical calculations do not suggest it is tending to zero.