This is the third paper in a series of four by the author et al. on algebraic methods for the study of spectral controllability and finite spectrum assignability for delay systems with finitely many commensurate delays and scalar control [see the author, {\it M. Ito} and {\it M. Kaneko}, ibid. 38, 913-926 (1983;

Zbl 0528.93037); the author and {\it M. Ito}, ibid. 39, 363-374 (1984;

Zbl 0531.93014) (not ibid. 38 (1983), 621, as it appears in the references of the paper under review); the author, {\it M. Ito} and {\it M. Kaneko}, ibid. 39, 1073-1082 (1984;

Zbl 0534.93045)]. The present paper proves necessity of a condition for spectral controllability in terms of finite Laplace transforms. Sufficiency has been proved earlier.