The suitability of different combinations of property data among vapor-pressure (P-sat), enthalpy of vaporization (Delta H-vap) and saturated-liquid heat-capacity (c(P,liq)(sat)) data for fitting consistent Twu alpha-function parameters is evaluated. A given combination is declared to be suitable if the obtained fitted parameters produce acceptable predictions for the three considered properties for the 783 molecules included in the reference database. Calculations are carried out using the translated-consistent versions of the Peng-Robinson (tc-PR) and Redlich-Kwong (tc-RK) cubic equations of state (CEoS). It is demonstrated that reliable parameters are obtained on the condition that they are, at least, fitted to vapor-pressure data. It is strongly advised to not exclusively fit alpha-function parameters to enthalpy of vaporization and/or saturated-liquid heat capacity data. Finally, Twu alpha-function parameters (L, M, N) suitable for the tc-PR and tc-RK CEoS are determined for the 1721 molecules, for which at least accurate vapor-pressure data are available in the DIPPR database. The volume-translation parameter c has also been determined but for only 1489 pure fluids. In addition, the generalized versions of the tc-PR and tc-RK CEoS ("generalized" means that the input parameters of these models are the experimental critical temperature, critical pressure, and acentric factor) are updated.