The Ho & Lee model for interest rates is given by the SDE: $$ \mathrm d r = \eta(t) \mathrm d t + c\,\mathrm d X $$ The calibration function for $\eta(t)$ is given by $$ \eta^*(t)=c^2(t-t^*)-\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t)) $$ where $Z_M(t^*, t)$ are the discount factors in the market from today $= t^*$ to maturity $t$ (Source: Paul Wilmott on Quantitative Finance, p. 526).
The term $\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t))$ confuses me.
I have a set of discount factors $Z_M$, which are numbers (e.g. $Z_M(0;\,0.5)=0.99750, Z_M(0;\,1)=0.989060)$.
So, the $\operatorname{log}Z_M$ is also a number.
How can I compute the partial derivative $\frac{\partial^2}{\partial t^2}\operatorname{log}(Z_M(t^*;t))$ of a number?
EDIT: My current understanding is that I have to use some interpolation method which is twice differentiable (for example spline interpolation) using the discount factors as support points. Would this be correct?