  Spectra of real alternating operators seem to be quite interesting from the
view point of explaining the Riemann Hypothesis for various zeta functions.
Unfortunately we have not sufficient experiments concerning this theme.
Necessary works would be to supply new examples of spectra related to zeros and
poles of zeta functions. A century ago Hilbert (1907) considered a kind of
operators representing quadratic forms of infinitely many variables.
Demonstrating the calculation of spectra for alternating Hilbert operators we
hope to present a novel scheme in this paper. Authors expect this study
encourages experts for further studies.
