Year | n | Author |
1903 | 15 | J. P. Gram [6] |
1914 | 79 | R. J. Backlund [1] |
1925 | 138 | J. I. Hutchinson [7] |
1935 | 1,041 | E. C. Titchmarsh [22] |
1953 | 1,104 | A. M. Turing [24] |
1956 | 15,000 | D. H. Lehmer [12] |
1956 | 25,000 | D. H. Lehmer [11] |
1958 | 35,337 | N. A. Meller [14] |
1966 | 250,000 | R. S. Lehman [10] |
1968 | 3,500,000 | J. B. Rosser, J. M. Yohe, L. Schoenfeld [21] |
1977 | 40,000,000 | R. P. Brent [2] |
1979 | 81,000,001 | R. P. Brent [3] |
1982 | 200,000,001 | R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter [25] |
1983 | 300,000,001 | J. van de Lune, H. J. J. te Riele [8] |
1986 | 1,500,000,001 | J. van de Lune, H. J. J. te Riele, D. T. Winter [9] |
2001 | 10,000,000,000 | J. van de Lune (unpublished) |
2004 | 900,000,000,000 | S. Wedeniwski [26] |
2004 | 10,000,000,000,000 | X. Gourdon and Patrick Demichel [5] |
|
| (1) |
|
Height | Total timing in hours | offset index of first zero | offset index of last zero | Value of R |
1013 | 33.1 | 1 | 2×109 | 16777216 |
1014 | 35.0 | 3 | 2×109 | 16777216 |
1015 | 38.3 | 0 | 2×109-1 | 8388608 |
1016 | 49.5 | 1 | 2×109-1 | 16777216 |
1017 | 46.9 | 0 | 2×109 | 16777216 |
1018 | 81.6 | 1 | 2×109-1 | 33554432 |
1019 | 65.9 | 0 | 2×109+1 | 33554432 |
1020 | 87.8 | 4 | 2×109-1 | 33554432 |
1021 | 139.9 | 0 | 2×109-1 | 33554432 |
1022 | 151.5 | 2 | 2×109-1 | 134217728 |
1023 | 219.0 | 100 | 2×109-1 | 268435456 |
1024 | 326.6 | 0 | 2×109+47 | 268435456 |
|
|
Height | VRR per million zeros | Number of types of VRR | Number of special points | Average number of points in VRR |
1013 | 37.98 | 68 | 282468 | 3.719 |
1014 | 54.10 | 86 | 418346 | 3.866 |
1015 | 72.42 | 109 | 581126 | 4.012 |
1016 | 93.99 | 140 | 780549 | 4.152 |
1017 | 117.25 | 171 | 1004269 | 4.283 |
1018 | 142.30 | 196 | 1255321 | 4.411 |
1019 | 168.55 | 225 | 1529685 | 4.538 |
1020 | 197.28 | 270 | 1837645 | 4.657 |
1021 | 225.80 | 322 | 2156944 | 4.776 |
1022 | 256.53 | 348 | 2507825 | 4.888 |
1023 | 286.97 | 480 | 2868206 | 4.997 |
1024 | 319.73 | 473 | 3262812 | 5.102 |
|
|
Height | Minimum of S(t) | Maximum of S(t) | Number of zeros with S(t) < -2.3 | Number of zeros with S(t) > 2.3 | Average number of change of sign of S(t) per Gram interval |
1013 | -2.4979 | 2.4775 | 208 | 237 | 1.5874 |
1014 | -2.5657 | 2.5822 | 481 | 411 | 1.5758 |
1015 | -2.7610 | 2.6318 | 785 | 760 | 1.5652 |
1016 | -2.6565 | 2.6094 | 1246 | 1189 | 1.5555 |
1017 | -2.6984 | 2.6961 | 1791 | 1812 | 1.5465 |
1018 | -2.8703 | 2.7141 | 2598 | 2743 | 1.5382 |
1019 | -2.9165 | 2.7553 | 3487 | 3467 | 1.5304 |
1020 | -2.7902 | 2.7916 | 4661 | 4603 | 1.5232 |
1021 | -2.7654 | 2.8220 | 5910 | 5777 | 1.5164 |
1022 | -2.8169 | 2.9796 | 7322 | 7359 | 1.5100 |
1023 | -2.8178 | 2.7989 | 8825 | 8898 | 1.5040 |
1024 | -2.9076 | 2.8799 | 10602 | 10598 | 1.4983 |
Height | Proportion of zeros computed twice | Mean difference for zeros computed twice | Max difference for zeros computed twice |
1013 | 4.0% | 5.90E-10 | 5.87E-07 |
1014 | 6.0% | 6.23E-10 | 1.43E-06 |
1015 | 6.0% | 7.81E-10 | 1.08E-06 |
1016 | 4.5% | 5.32E-10 | 7.75E-07 |
1017 | 8.0% | 5.85E-10 | 9.22E-07 |
1018 | 7.5% | 6.59E-10 | 1.88E-06 |
1019 | 11.0% | 5.15E-10 | 3.07E-06 |
1020 | 12.5% | 3.93E-10 | 7.00E-07 |
1021 | 31.5% | 5.64E-10 | 3.54E-06 |
1022 | 50.0% | 1.15E-09 | 2.39E-06 |
1023 | 50.0% | 1.34E-09 | 3.11E-06 |
1024 | 50.0% | 2.68E-09 | 6.82E-06 |
|
|
|
|
Height | Mini dn | Maxi dn | Mini dn+dn+1 | Maxi dn+dn+1 | Prob min dn in GUE | Prob min dn+dn+1 in GUE |
1013 | 0.0005330 | 4.127 | 0.1097 | 5.232 | 0.28 | 0.71 |
1014 | 0.0009764 | 4.236 | 0.1213 | 5.349 | 0.87 | 0.94 |
1015 | 0.0005171 | 4.154 | 0.1003 | 5.434 | 0.26 | 0.46 |
1016 | 0.0005202 | 4.202 | 0.1029 | 5.433 | 0.27 | 0.53 |
1017 | 0.0006583 | 4.183 | 0.0966 | 5.395 | 0.47 | 0.36 |
1018 | 0.0004390 | 4.194 | 0.1080 | 5.511 | 0.17 | 0.67 |
1019 | 0.0004969 | 4.200 | 0.0874 | 5.341 | 0.24 | 0.18 |
1020 | 0.0004351 | 4.268 | 0.1067 | 5.717 | 0.17 | 0.63 |
1021 | 0.0004934 | 4.316 | 0.1019 | 5.421 | 0.23 | 0.50 |
1022 | 0.0008161 | 4.347 | 0.1060 | 5.332 | 0.70 | 0.61 |
1023 | 0.0004249 | 4.304 | 0.1112 | 5.478 | 0.15 | 0.75 |
1024 | 0.0002799 | 4.158 | 0.0877 | 5.526 | 0.05 | 0.19 |
|
| (2) |
|
Height | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 |
1013 | 0.17608 | 0.03512 | 0.09608 | 0.05933 | 0.10107 | 0.1095 | 0.1719 | 0.2471 |
1014 | 0.17657 | 0.03540 | 0.09663 | 0.05990 | 0.10199 | 0.1108 | 0.1741 | 0.2510 |
1015 | 0.17697 | 0.03565 | 0.09710 | 0.06040 | 0.10277 | 0.1119 | 0.1759 | 0.2539 |
1016 | 0.17732 | 0.03586 | 0.09750 | 0.06084 | 0.10347 | 0.1129 | 0.1776 | 0.2567 |
1017 | 0.17760 | 0.03605 | 0.09785 | 0.06123 | 0.10407 | 0.1137 | 0.1789 | 0.2590 |
1018 | 0.17784 | 0.03621 | 0.09816 | 0.06157 | 0.10462 | 0.1145 | 0.1803 | 0.2613 |
1019 | 0.17805 | 0.03636 | 0.09843 | 0.06189 | 0.10511 | 0.1152 | 0.1814 | 0.2631 |
1020 | 0.17824 | 0.03649 | 0.09867 | 0.06215 | 0.10553 | 0.1158 | 0.1824 | 0.2649 |
1021 | 0.17839 | 0.03661 | 0.09888 | 0.06242 | 0.10595 | 0.1165 | 0.1836 | 0.2668 |
1022 | 0.17853 | 0.03671 | 0.09906 | 0.06262 | 0.10627 | 0.1169 | 0.1842 | 0.2678 |
1023 | 0.17864 | 0.03680 | 0.09922 | 0.06282 | 0.10658 | 0.1174 | 0.1850 | 0.2692 |
1024 | 0.17875 | 0.03688 | 0.09937 | 0.06301 | 0.10689 | 0.1179 | 0.1859 | 0.2708 |
GUE | 0.17999 | 0.03796 | 0.10130 | 0.06552 | 0.11096 | 0.1243 | 0.1969 | 0.2902 |
|
Height | N2 | N3 | N4 | N5 | N6 | N7 | N8 | N9 |
1013 | 0.23717 | 0.02671 | 0.16887 | 0.06252 | 0.2073 | 0.1530 | 0.3764 | 0.4304 |
1014 | 0.23846 | 0.02678 | 0.17045 | 0.06301 | 0.2099 | 0.1550 | 0.3827 | 0.4388 |
1015 | 0.23956 | 0.02688 | 0.17181 | 0.06349 | 0.2122 | 0.1568 | 0.3880 | 0.4458 |
1016 | 0.24050 | 0.02700 | 0.17299 | 0.06396 | 0.2142 | 0.1585 | 0.3927 | 0.4523 |
1017 | 0.24132 | 0.02713 | 0.17404 | 0.06446 | 0.2159 | 0.1601 | 0.3970 | 0.4583 |
1018 | 0.24202 | 0.02726 | 0.17494 | 0.06488 | 0.2175 | 0.1614 | 0.4005 | 0.4630 |
1019 | 0.24264 | 0.02740 | 0.17574 | 0.06530 | 0.2188 | 0.1627 | 0.4036 | 0.4672 |
1020 | 0.24319 | 0.02753 | 0.17645 | 0.06569 | 0.2201 | 0.1639 | 0.4065 | 0.4713 |
1021 | 0.24366 | 0.02766 | 0.17709 | 0.06609 | 0.2212 | 0.1651 | 0.4092 | 0.4753 |
1022 | 0.24409 | 0.02778 | 0.17765 | 0.06643 | 0.2222 | 0.1660 | 0.4114 | 0.4780 |
1023 | 0.24447 | 0.02790 | 0.17819 | 0.06679 | 0.2232 | 0.1671 | 0.4140 | 0.4821 |
1024 | 0.24480 | 0.02801 | 0.17863 | 0.06709 | 0.2240 | 0.1679 | 0.4158 | 0.4846 |
GUE | 0.249 | 0.03 | 0.185 | 0.073 | 0.237 | 0.185 | 0.451 | 0.544 |
Height | logdn | 1/dn | 1/dn2 |
1013 | -0.101540 | 1.27050 | 2.52688 |
1014 | -0.101798 | 1.27124 | 2.53173 |
1015 | -0.102009 | 1.27184 | 2.54068 |
1016 | -0.102188 | 1.27235 | 2.54068 |
1017 | -0.102329 | 1.27272 | 2.54049 |
1018 | -0.102453 | 1.27308 | 2.54540 |
1019 | -0.102558 | 1.27338 | 2.54906 |
1020 | -0.102650 | 1.27363 | 2.54996 |
1021 | -0.102721 | 1.27382 | 2.54990 |
1022 | -0.102789 | 1.27401 | 2.54783 |
1023 | -0.102843 | 1.27415 | 2.55166 |
1024 | -0.102891 | 1.27427 | 2.55728 |
GUE | -0.1035 | 1.2758 | 2.5633 |