The object of the following paper is to simplify the methods and formulas developed and used by Gram,t Lindelöf,^ and Backlund, § in numerical investigations connected with the roots of the Zeta function. I apply these to locating and calculating additional roots. I start with the formulas, as given by Backlund,

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Two linearly independent integrals are fdx f(x-ßx)(x-ß2)dx Mi=Jy M'=J-?-• If for a particular value of x the corresponding values of y in the first, second, and third sheets of the surface are denoted by yx, y2, y3 respectively, we shall assume y2 = pyx, y3 = p2y2, where p = J(-1 + \/-3~). Denoting the moduli of periodicity of the integrals u" at the cuts a,, a2, bx, b2 by A*x, Av2, BvX, Bv2 respectively, one sees easily from the above figure that Ar2=-p"AvX, Bv2 = p*>BvX, and derives the following table of periods.

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