Get the discrete-time poles and zeros from continuous-time poles and zeros

How do you implement the following function:

$$[Z^d, P^d, K^d] = text{fcn} ,(Z^c, P^c, K^c),$$

where $Z^c = [z^c_m, …, z^c_1]$, $P^c = [p^c_m, …, p^c_1]$, and $K^c$ are zeros, poles, and gain of the continuous-time TF, and similarly $Z^d$, $P^d$, and $K^d$ are zeros, poles, and gain of the equivalent discrete-time TF with sampling time of $T_s$.

I have tried the following but it did not give me the correct answer:

$Z^d = e^{Z^c times T_s}$

$P^d = e^{P^c times T_s}$

$K^d = 1-e^{-K^c times T_s}$ —> (is the minus sign right?)

I am looking for some conversion formula that can handle all cases (including single/multiple poles or zeros).

Note: I do not want to use any Matlab built-in functions like c2d(), zp2tf(), or tf().