Gram point is a solution of These solutions are approximated by the sequence: where is the Lambert W function. Here are the smallest non negative Gram points The choice of the index n is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this -function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [−24,24]! Thus the odd theta-function has its symmetric Gram point with value 0 at index −3. Gram points are useful when computing the zeros of . At a Gram point and if this is positive at two successive Gram points, must have a zero in the interval. According to Gram’s law, the real part is usually positive while the imaginary part alternates with the Gram points, between positive and negative values at somewhat regular intervals. The number of roots, , in the strip from 0 to T, can be found by where is an error term which grows asymptotically like . Only if would obey Gram’s law, then finding the number of roots in the strip simply becomes Today we know, that in the long run, Gram's law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase Gram's law for the (false) statement that all zeroes on the critical line would be separated by Gram points. See also Z function References Gabcke, W. (1979), Neue Herleitung und explizierte Restabschätzung der Riemann-Siegel-Formel. Thesis, University of Göttingen. Revised version (eDiss Göttingen 2015) External links Wolfram Research – Riemann-Siegel Theta function (includes function plotting and evaluation) Zeta and L-functions Bernhard Riemann