The zeta-function of Fibonacci sequence?

Question

I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some familiar examples to start with. But I failed to tell what the zeta-function for the Fibonacci sequence is.
WHat I have tried so far is: By use of the closed-form, we might tell that $N_r=\frac{(1 + \sqrt{5})^r - (1 - \sqrt{5})^r}{2^r \sqrt{5}}$. Hence, by means of logarithms, we can derive that the function ought to be equal to ${(1-(1 - \sqrt{5})T/2)/[(1-(1 + \sqrt{5})T/2)]}^{1/ \sqrt{5}}$. But this formula seems to be just to... hard to accept. So I suspect that I must be wrong somewhere. Then am I wrong? And are there other proofs of this relation? And, in general, how to derive the zeta-function for a given sequence? Thanks in advance.