Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$.
Consider a state $\omega$.
Does it suffice to have on a dense set the KMS-condition: $$F(t+i\beta)=\lim_nF_n(t+i\beta)=\omega(\tau^t[B_n]A_n)=\omega(\tau^t[B]A)$$ (I wonder wether it may fail to be analytic; is this important?)
(Usually, one can check this for the dense set of entire elements.)
No, it does not suffice!
It can even severely fail to extend to the complex plane at all.
(See the thread: Analyticity: Uniform Limit)