استعمل برنارد ريمان صيغةريمان-سيغل (دالة لم تنشر ولكنا ذُكرت في harvnb سيغل 1932 ).
J. P. harvtxt غرام 1903 استعمل صيغة أويلر-ماكلورين فاكتشفقانون غرام. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he found.
79 (خ³< >n > â‰¤ 200)
R. J. harvtxt Backlund 1914 introduced a better method of checking all the zeros up to that point are on the line, by studying the argument < >S >(< >T >) of the zeta function.
138 (خ³< >n > â‰¤ 300)
J. I. harvtxt Hutchinson 1925 found the first failure of Gram's law, at the Gram point < >g >126.
E. C. harvtxt Titchmarsh 1935 used the recently rediscovered صيغةريمان-سيغل , which is much faster than Euler–Maclaurin summation.It takes about O(< >T >3/2+خµ) steps to check zeros with imaginary part less than < >T >, while the Euler–Maclaurin method takes about O(< >T >2+خµ) steps.
E. C. harvtxt Titchmarsh 1936 and L. J. Comrie were the last to find zeros by hand.
A. M. harvtxt Turing 1953 found a more efficient way to check that all zeros up to some point are accounted for by the zeros on the line, by checking that < >Z > has the correct sign at several consecutive Gram points and using the fact that < >S >(< >T >) has average value 0. This requires almost no extra work because the sign of < >Z > at Gram points is already known from finding the zeros, and is still the usual method used. This was the first use of a digital computer to calculate the zeros.
gaps 15 000
D. H. harvtxt Lehmer 1956 discovered a few cases where the zeta function has zeros that are only just on the line two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between th . This is called Lehmer's phenomenon , and first occurs at the zeros with imaginary parts 7005.063 and 7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1.
R. P. Brent, Johan van de Lune J. van de Lune , Herman te Riele H. J. J. te Riele , D. T. Winter
gaps 300 000 001
J. van de Lune, H. J. J. te Riele
gaps 1 500 000 001
harvtxt van de Lune te Riele Winter 1986 gave some statistical data about the zeros and give several graphs of < >Z > at places where it has unusual behavior.
A few of large (~1012) height
harvs txt first A. M. last Odlyzko year1 1987 computed smaller numbers of zeros of much larger height, around 1012, to high precision to check Montgomery's pair correlation conjecture .
A few of large (~1020) height
harvs txt first A. M. last Odlyzko year1 1992 computed a 175 million zeroes of heights around 1020 and a few more of heights around 2 e 20 , and gave an extensive discussion of the results.
10000 of large (~1021) height
harvs txt first A. M. last Odlyzko year1 1998 computed some zeros of height about 1021
gaps 10 000 000 000
J. van de Lune (unpublished)
gaps 900 000 000 000
S. Wedeniwski ( ZetaGrid distributed computing)
gaps 10 000 000 000 000 and a few of large (up to ~1024) heights
X. harvtxt Gourdon 2004 and Patrick D ichel used the Odlyzko–Schأ¶nhage algorithm . They also checked two billion zeros around heights 1013, 1014, ... , 1024.