Dirichlet series
| L(s) = 1 | − 6·2-s − 2·3-s + 11·4-s + 12·6-s + 2·7-s + 6·8-s − 4·9-s − 8·11-s − 22·12-s − 12·14-s − 45·16-s + 10·17-s + 24·18-s − 16·19-s − 4·21-s + 48·22-s + 2·23-s − 12·24-s + 9·25-s + 16·27-s + 22·28-s + 30·32-s + 16·33-s − 60·34-s − 44·36-s + 4·37-s + 96·38-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 1.15·3-s + 11/2·4-s + 4.89·6-s + 0.755·7-s + 2.12·8-s − 4/3·9-s − 2.41·11-s − 6.35·12-s − 3.20·14-s − 11.2·16-s + 2.42·17-s + 5.65·18-s − 3.67·19-s − 0.872·21-s + 10.2·22-s + 0.417·23-s − 2.44·24-s + 9/5·25-s + 3.07·27-s + 4.15·28-s + 5.30·32-s + 2.78·33-s − 10.2·34-s − 7.33·36-s + 0.657·37-s + 15.5·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 13^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.82539\times 10^{16}\) |
| Root analytic conductor: | \(2.59756\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{20} \cdot 13^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.06918921894\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06918921894\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 9 T^{2} + 33 T^{4} - 16 T^{5} - 24 T^{6} + 48 T^{7} - 906 T^{8} + 96 T^{9} + 7346 T^{10} + 96 p T^{11} - 906 p^{2} T^{12} + 48 p^{3} T^{13} - 24 p^{4} T^{14} - 16 p^{5} T^{15} + 33 p^{6} T^{16} - 9 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T + 25 T^{2} + 39 p T^{3} + 101 p T^{4} + 111 p^{2} T^{5} + 845 T^{6} + 177 p^{3} T^{7} + 2105 T^{8} + 357 p^{3} T^{9} + 29 p^{7} T^{10} + 159 p^{5} T^{11} + 7953 T^{12} + 6861 p T^{13} + 23753 T^{14} + 18951 p T^{15} + 26915 p T^{16} + 261 p^{8} T^{17} + 73281 T^{18} + 9567 p^{3} T^{19} + 92661 T^{20} + 9567 p^{4} T^{21} + 73281 p^{2} T^{22} + 261 p^{11} T^{23} + 26915 p^{5} T^{24} + 18951 p^{6} T^{25} + 23753 p^{6} T^{26} + 6861 p^{8} T^{27} + 7953 p^{8} T^{28} + 159 p^{14} T^{29} + 29 p^{17} T^{30} + 357 p^{14} T^{31} + 2105 p^{12} T^{32} + 177 p^{16} T^{33} + 845 p^{14} T^{34} + 111 p^{17} T^{35} + 101 p^{17} T^{36} + 39 p^{18} T^{37} + 25 p^{18} T^{38} + 3 p^{20} T^{39} + p^{20} T^{40} \) |
| 3 | \( 1 + 2 T + 8 T^{2} + 8 T^{3} + 25 T^{4} + 20 T^{5} + 76 T^{6} + 14 p^{2} T^{7} + 352 T^{8} + 730 T^{9} + 1348 T^{10} + 2660 T^{11} + 4147 T^{12} + 316 p^{3} T^{13} + 16520 T^{14} + 33422 T^{15} + 66395 T^{16} + 99232 T^{17} + 64880 p T^{18} + 78548 p T^{19} + 540160 T^{20} + 78548 p^{2} T^{21} + 64880 p^{3} T^{22} + 99232 p^{3} T^{23} + 66395 p^{4} T^{24} + 33422 p^{5} T^{25} + 16520 p^{6} T^{26} + 316 p^{10} T^{27} + 4147 p^{8} T^{28} + 2660 p^{9} T^{29} + 1348 p^{10} T^{30} + 730 p^{11} T^{31} + 352 p^{12} T^{32} + 14 p^{15} T^{33} + 76 p^{14} T^{34} + 20 p^{15} T^{35} + 25 p^{16} T^{36} + 8 p^{17} T^{37} + 8 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 2 T - 6 p T^{2} + 68 T^{3} + 929 T^{4} - 1124 T^{5} - 14194 T^{6} + 1598 p T^{7} + 168832 T^{8} - 9642 p T^{9} - 1684962 T^{10} + 185940 T^{11} + 2130957 p T^{12} + 472932 T^{13} - 121949582 T^{14} - 5345154 T^{15} + 943522539 T^{16} + 7858360 T^{17} - 6992208836 T^{18} + 30581128 T^{19} + 49860115936 T^{20} + 30581128 p T^{21} - 6992208836 p^{2} T^{22} + 7858360 p^{3} T^{23} + 943522539 p^{4} T^{24} - 5345154 p^{5} T^{25} - 121949582 p^{6} T^{26} + 472932 p^{7} T^{27} + 2130957 p^{9} T^{28} + 185940 p^{9} T^{29} - 1684962 p^{10} T^{30} - 9642 p^{12} T^{31} + 168832 p^{12} T^{32} + 1598 p^{14} T^{33} - 14194 p^{14} T^{34} - 1124 p^{15} T^{35} + 929 p^{16} T^{36} + 68 p^{17} T^{37} - 6 p^{19} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 8 T + 20 T^{2} - 24 T^{3} - 219 T^{4} - 888 T^{5} - 4284 T^{6} - 15880 T^{7} - 12564 T^{8} + 158360 T^{9} + 899332 T^{10} + 2756696 T^{11} + 7669307 T^{12} + 25336 p^{2} T^{13} - 105036036 T^{14} - 513543080 T^{15} - 1219479681 T^{16} - 2337775184 T^{17} + 1553351912 T^{18} + 44546706496 T^{19} + 220101622920 T^{20} + 44546706496 p T^{21} + 1553351912 p^{2} T^{22} - 2337775184 p^{3} T^{23} - 1219479681 p^{4} T^{24} - 513543080 p^{5} T^{25} - 105036036 p^{6} T^{26} + 25336 p^{9} T^{27} + 7669307 p^{8} T^{28} + 2756696 p^{9} T^{29} + 899332 p^{10} T^{30} + 158360 p^{11} T^{31} - 12564 p^{12} T^{32} - 15880 p^{13} T^{33} - 4284 p^{14} T^{34} - 888 p^{15} T^{35} - 219 p^{16} T^{36} - 24 p^{17} T^{37} + 20 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 10 T - 7 T^{2} + 470 T^{3} - 1882 T^{4} - 2042 T^{5} + 40575 T^{6} - 177466 T^{7} + 251190 T^{8} + 2495842 T^{9} - 12749509 T^{10} - 18741814 T^{11} + 217405948 T^{12} + 251724858 T^{13} - 4973586989 T^{14} + 12867053474 T^{15} + 23809984513 T^{16} - 417918910624 T^{17} + 1648824925642 T^{18} + 3203645045440 T^{19} - 42570382777428 T^{20} + 3203645045440 p T^{21} + 1648824925642 p^{2} T^{22} - 417918910624 p^{3} T^{23} + 23809984513 p^{4} T^{24} + 12867053474 p^{5} T^{25} - 4973586989 p^{6} T^{26} + 251724858 p^{7} T^{27} + 217405948 p^{8} T^{28} - 18741814 p^{9} T^{29} - 12749509 p^{10} T^{30} + 2495842 p^{11} T^{31} + 251190 p^{12} T^{32} - 177466 p^{13} T^{33} + 40575 p^{14} T^{34} - 2042 p^{15} T^{35} - 1882 p^{16} T^{36} + 470 p^{17} T^{37} - 7 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 16 T + 176 T^{2} + 1460 T^{3} + 10321 T^{4} + 63732 T^{5} + 356664 T^{6} + 1829664 T^{7} + 8759124 T^{8} + 39787552 T^{9} + 174977008 T^{10} + 761420092 T^{11} + 3335898519 T^{12} + 14694366620 T^{13} + 64786527696 T^{14} + 282753607560 T^{15} + 1213406803719 T^{16} + 5111934152664 T^{17} + 21227495686040 T^{18} + 88894314395488 T^{19} + 379894945670616 T^{20} + 88894314395488 p T^{21} + 21227495686040 p^{2} T^{22} + 5111934152664 p^{3} T^{23} + 1213406803719 p^{4} T^{24} + 282753607560 p^{5} T^{25} + 64786527696 p^{6} T^{26} + 14694366620 p^{7} T^{27} + 3335898519 p^{8} T^{28} + 761420092 p^{9} T^{29} + 174977008 p^{10} T^{30} + 39787552 p^{11} T^{31} + 8759124 p^{12} T^{32} + 1829664 p^{13} T^{33} + 356664 p^{14} T^{34} + 63732 p^{15} T^{35} + 10321 p^{16} T^{36} + 1460 p^{17} T^{37} + 176 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 2 T - 52 T^{2} + 160 T^{3} + 157 T^{4} - 6596 T^{5} + 46576 T^{6} + 124218 T^{7} - 1185616 T^{8} + 2300286 T^{9} + 6454608 T^{10} - 158042596 T^{11} + 339796655 T^{12} + 40412020 p T^{13} - 16319998268 T^{14} + 85172481666 T^{15} + 252762307659 T^{16} - 1967344793632 T^{17} + 8221712690560 T^{18} + 12847528296684 T^{19} - 399858827628576 T^{20} + 12847528296684 p T^{21} + 8221712690560 p^{2} T^{22} - 1967344793632 p^{3} T^{23} + 252762307659 p^{4} T^{24} + 85172481666 p^{5} T^{25} - 16319998268 p^{6} T^{26} + 40412020 p^{8} T^{27} + 339796655 p^{8} T^{28} - 158042596 p^{9} T^{29} + 6454608 p^{10} T^{30} + 2300286 p^{11} T^{31} - 1185616 p^{12} T^{32} + 124218 p^{13} T^{33} + 46576 p^{14} T^{34} - 6596 p^{15} T^{35} + 157 p^{16} T^{36} + 160 p^{17} T^{37} - 52 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 117 T^{2} + 6870 T^{4} + 8328 T^{5} + 9015 p T^{6} + 908016 T^{7} + 6368106 T^{8} + 49449240 T^{9} + 86554363 T^{10} + 1654451928 T^{11} - 517414740 T^{12} + 27891514632 T^{13} - 83953593117 T^{14} - 350011028904 T^{15} - 3978002845539 T^{16} - 1543536257664 p T^{17} - 154268740846974 T^{18} - 1909124569268712 T^{19} - 4952841504137748 T^{20} - 1909124569268712 p T^{21} - 154268740846974 p^{2} T^{22} - 1543536257664 p^{4} T^{23} - 3978002845539 p^{4} T^{24} - 350011028904 p^{5} T^{25} - 83953593117 p^{6} T^{26} + 27891514632 p^{7} T^{27} - 517414740 p^{8} T^{28} + 1654451928 p^{9} T^{29} + 86554363 p^{10} T^{30} + 49449240 p^{11} T^{31} + 6368106 p^{12} T^{32} + 908016 p^{13} T^{33} + 9015 p^{15} T^{34} + 8328 p^{15} T^{35} + 6870 p^{16} T^{36} + 117 p^{18} T^{38} + p^{20} T^{40} \) | |
| 31 | \( 1 + 104 T^{3} + 1794 T^{4} + 3320 T^{5} + 5408 T^{6} + 552640 T^{7} + 1315037 T^{8} - 1490240 T^{9} + 53283808 T^{10} + 779482432 T^{11} + 2015402168 T^{12} - 9179164672 T^{13} + 126121189280 T^{14} + 557942017984 T^{15} - 69210310750 T^{16} + 4465408437184 T^{17} + 158380681056 p^{2} T^{18} + 15988767100848 p T^{19} - 2929717194662260 T^{20} + 15988767100848 p^{2} T^{21} + 158380681056 p^{4} T^{22} + 4465408437184 p^{3} T^{23} - 69210310750 p^{4} T^{24} + 557942017984 p^{5} T^{25} + 126121189280 p^{6} T^{26} - 9179164672 p^{7} T^{27} + 2015402168 p^{8} T^{28} + 779482432 p^{9} T^{29} + 53283808 p^{10} T^{30} - 1490240 p^{11} T^{31} + 1315037 p^{12} T^{32} + 552640 p^{13} T^{33} + 5408 p^{14} T^{34} + 3320 p^{15} T^{35} + 1794 p^{16} T^{36} + 104 p^{17} T^{37} + p^{20} T^{40} \) | |
| 37 | \( 1 - 4 T - 255 T^{2} + 628 T^{3} + 35070 T^{4} - 42996 T^{5} - 3363009 T^{6} + 1430592 T^{7} + 250169718 T^{8} + 2742320 T^{9} - 15589359357 T^{10} - 2630094216 T^{11} + 857793078820 T^{12} + 140911146544 T^{13} - 42819935288493 T^{14} - 4890972627376 T^{15} + 1957545276095489 T^{16} + 128213364289956 T^{17} - 82094063860519094 T^{18} - 1723066089909848 T^{19} + 3164986030262490540 T^{20} - 1723066089909848 p T^{21} - 82094063860519094 p^{2} T^{22} + 128213364289956 p^{3} T^{23} + 1957545276095489 p^{4} T^{24} - 4890972627376 p^{5} T^{25} - 42819935288493 p^{6} T^{26} + 140911146544 p^{7} T^{27} + 857793078820 p^{8} T^{28} - 2630094216 p^{9} T^{29} - 15589359357 p^{10} T^{30} + 2742320 p^{11} T^{31} + 250169718 p^{12} T^{32} + 1430592 p^{13} T^{33} - 3363009 p^{14} T^{34} - 42996 p^{15} T^{35} + 35070 p^{16} T^{36} + 628 p^{17} T^{37} - 255 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 28 T + 389 T^{2} + 3008 T^{3} + 8194 T^{4} - 92960 T^{5} - 1264061 T^{6} - 6519924 T^{7} - 3613294 T^{8} + 159299460 T^{9} + 822628347 T^{10} - 787403984 T^{11} - 19305288280 T^{12} + 60662487584 T^{13} + 2225226104059 T^{14} + 20587364409156 T^{15} + 114095998027205 T^{16} + 292309852251568 T^{17} - 1972676048100766 T^{18} - 34827701209580176 T^{19} - 281355382830851844 T^{20} - 34827701209580176 p T^{21} - 1972676048100766 p^{2} T^{22} + 292309852251568 p^{3} T^{23} + 114095998027205 p^{4} T^{24} + 20587364409156 p^{5} T^{25} + 2225226104059 p^{6} T^{26} + 60662487584 p^{7} T^{27} - 19305288280 p^{8} T^{28} - 787403984 p^{9} T^{29} + 822628347 p^{10} T^{30} + 159299460 p^{11} T^{31} - 3613294 p^{12} T^{32} - 6519924 p^{13} T^{33} - 1264061 p^{14} T^{34} - 92960 p^{15} T^{35} + 8194 p^{16} T^{36} + 3008 p^{17} T^{37} + 389 p^{18} T^{38} + 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 22 T + 332 T^{2} - 3124 T^{3} + 24073 T^{4} - 137704 T^{5} + 827200 T^{6} - 115270 p T^{7} + 40616992 T^{8} - 296972046 T^{9} + 2336560152 T^{10} - 16772667664 T^{11} + 143141988371 T^{12} - 1164757348680 T^{13} + 9499206282004 T^{14} - 64461164121898 T^{15} + 415971606249867 T^{16} - 2327476919488472 T^{17} + 14227460893269592 T^{18} - 81794197398086924 T^{19} + 556966076767085568 T^{20} - 81794197398086924 p T^{21} + 14227460893269592 p^{2} T^{22} - 2327476919488472 p^{3} T^{23} + 415971606249867 p^{4} T^{24} - 64461164121898 p^{5} T^{25} + 9499206282004 p^{6} T^{26} - 1164757348680 p^{7} T^{27} + 143141988371 p^{8} T^{28} - 16772667664 p^{9} T^{29} + 2336560152 p^{10} T^{30} - 296972046 p^{11} T^{31} + 40616992 p^{12} T^{32} - 115270 p^{14} T^{33} + 827200 p^{14} T^{34} - 137704 p^{15} T^{35} + 24073 p^{16} T^{36} - 3124 p^{17} T^{37} + 332 p^{18} T^{38} - 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( ( 1 - 20 T + 486 T^{2} - 6548 T^{3} + 93533 T^{4} - 960312 T^{5} + 10222280 T^{6} - 85662136 T^{7} + 747535234 T^{8} - 5352082240 T^{9} + 40243643940 T^{10} - 5352082240 p T^{11} + 747535234 p^{2} T^{12} - 85662136 p^{3} T^{13} + 10222280 p^{4} T^{14} - 960312 p^{5} T^{15} + 93533 p^{6} T^{16} - 6548 p^{7} T^{17} + 486 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 + 10 T + 50 T^{2} + 608 T^{3} + 8911 T^{4} + 42256 T^{5} + 161842 T^{6} + 1379418 T^{7} + 11864269 T^{8} + 42078400 T^{9} + 116867400 T^{10} - 342809968 T^{11} + 2993373476 T^{12} + 82242842224 T^{13} + 232250742760 T^{14} - 4408284842096 T^{15} - 104445179440542 T^{16} - 7400825808492 p T^{17} - 1433330906377060 T^{18} - 34029241052501104 T^{19} - 545288959843663782 T^{20} - 34029241052501104 p T^{21} - 1433330906377060 p^{2} T^{22} - 7400825808492 p^{4} T^{23} - 104445179440542 p^{4} T^{24} - 4408284842096 p^{5} T^{25} + 232250742760 p^{6} T^{26} + 82242842224 p^{7} T^{27} + 2993373476 p^{8} T^{28} - 342809968 p^{9} T^{29} + 116867400 p^{10} T^{30} + 42078400 p^{11} T^{31} + 11864269 p^{12} T^{32} + 1379418 p^{13} T^{33} + 161842 p^{14} T^{34} + 42256 p^{15} T^{35} + 8911 p^{16} T^{36} + 608 p^{17} T^{37} + 50 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 8 T - 160 T^{2} - 1360 T^{3} + 6013 T^{4} + 102072 T^{5} + 738528 T^{6} - 2533208 T^{7} - 73281532 T^{8} - 209404528 T^{9} + 37464416 T^{10} + 26337573400 T^{11} + 280444882123 T^{12} - 1108551577128 T^{13} - 11218760651904 T^{14} - 24021446097136 T^{15} - 143672083299481 T^{16} + 5273614558649664 T^{17} + 5977156174031072 T^{18} - 175413463692198584 T^{19} + 707142863345228216 T^{20} - 175413463692198584 p T^{21} + 5977156174031072 p^{2} T^{22} + 5273614558649664 p^{3} T^{23} - 143672083299481 p^{4} T^{24} - 24021446097136 p^{5} T^{25} - 11218760651904 p^{6} T^{26} - 1108551577128 p^{7} T^{27} + 280444882123 p^{8} T^{28} + 26337573400 p^{9} T^{29} + 37464416 p^{10} T^{30} - 209404528 p^{11} T^{31} - 73281532 p^{12} T^{32} - 2533208 p^{13} T^{33} + 738528 p^{14} T^{34} + 102072 p^{15} T^{35} + 6013 p^{16} T^{36} - 1360 p^{17} T^{37} - 160 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 16 T - 291 T^{2} - 4624 T^{3} + 62614 T^{4} + 791392 T^{5} - 10200029 T^{6} - 91274480 T^{7} + 1347937546 T^{8} + 7746824480 T^{9} - 143223609453 T^{10} - 467039401632 T^{11} + 12554800039532 T^{12} + 18247567668448 T^{13} - 918360371027813 T^{14} - 202692877277696 T^{15} + 58440043341660413 T^{16} - 20244855642251520 T^{17} - 3426930072683677342 T^{18} + 853407459595356016 T^{19} + \)\(20\!\cdots\!68\)\( T^{20} + 853407459595356016 p T^{21} - 3426930072683677342 p^{2} T^{22} - 20244855642251520 p^{3} T^{23} + 58440043341660413 p^{4} T^{24} - 202692877277696 p^{5} T^{25} - 918360371027813 p^{6} T^{26} + 18247567668448 p^{7} T^{27} + 12554800039532 p^{8} T^{28} - 467039401632 p^{9} T^{29} - 143223609453 p^{10} T^{30} + 7746824480 p^{11} T^{31} + 1347937546 p^{12} T^{32} - 91274480 p^{13} T^{33} - 10200029 p^{14} T^{34} + 791392 p^{15} T^{35} + 62614 p^{16} T^{36} - 4624 p^{17} T^{37} - 291 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 18 T + 470 T^{2} - 6516 T^{3} + 104201 T^{4} - 1121100 T^{5} + 13431934 T^{6} - 112895598 T^{7} + 1056770664 T^{8} - 6211198182 T^{9} + 42069621398 T^{10} - 40505100180 T^{11} - 508608587109 T^{12} + 19726130422932 T^{13} - 115190196069214 T^{14} + 761930116963614 T^{15} + 6287387668895859 T^{16} - 128665633206649848 T^{17} + 2026059069895604484 T^{18} - 20095962770938094832 T^{19} + \)\(18\!\cdots\!44\)\( T^{20} - 20095962770938094832 p T^{21} + 2026059069895604484 p^{2} T^{22} - 128665633206649848 p^{3} T^{23} + 6287387668895859 p^{4} T^{24} + 761930116963614 p^{5} T^{25} - 115190196069214 p^{6} T^{26} + 19726130422932 p^{7} T^{27} - 508608587109 p^{8} T^{28} - 40505100180 p^{9} T^{29} + 42069621398 p^{10} T^{30} - 6211198182 p^{11} T^{31} + 1056770664 p^{12} T^{32} - 112895598 p^{13} T^{33} + 13431934 p^{14} T^{34} - 1121100 p^{15} T^{35} + 104201 p^{16} T^{36} - 6516 p^{17} T^{37} + 470 p^{18} T^{38} - 18 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 56 T + 1952 T^{2} + 48396 T^{3} + 953561 T^{4} + 15523292 T^{5} + 217618632 T^{6} + 2711712552 T^{7} + 31230772164 T^{8} + 345744549544 T^{9} + 3784871417104 T^{10} + 41211074532692 T^{11} + 438822748561071 T^{12} + 4488068016117828 T^{13} + 43707379067808384 T^{14} + 408784301786439440 T^{15} + 3737075509158186519 T^{16} + 33940694818139883288 T^{17} + \)\(30\!\cdots\!88\)\( T^{18} + \)\(27\!\cdots\!52\)\( T^{19} + \)\(23\!\cdots\!44\)\( T^{20} + \)\(27\!\cdots\!52\)\( p T^{21} + \)\(30\!\cdots\!88\)\( p^{2} T^{22} + 33940694818139883288 p^{3} T^{23} + 3737075509158186519 p^{4} T^{24} + 408784301786439440 p^{5} T^{25} + 43707379067808384 p^{6} T^{26} + 4488068016117828 p^{7} T^{27} + 438822748561071 p^{8} T^{28} + 41211074532692 p^{9} T^{29} + 3784871417104 p^{10} T^{30} + 345744549544 p^{11} T^{31} + 31230772164 p^{12} T^{32} + 2711712552 p^{13} T^{33} + 217618632 p^{14} T^{34} + 15523292 p^{15} T^{35} + 953561 p^{16} T^{36} + 48396 p^{17} T^{37} + 1952 p^{18} T^{38} + 56 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 662 T^{2} + 221179 T^{4} - 49439326 T^{6} + 8334146485 T^{8} - 1138166350304 T^{10} + 132447463647444 T^{12} - 13610776696011280 T^{14} + 1261445493668289578 T^{16} - \)\(10\!\cdots\!72\)\( T^{18} + \)\(81\!\cdots\!98\)\( T^{20} - \)\(10\!\cdots\!72\)\( p^{2} T^{22} + 1261445493668289578 p^{4} T^{24} - 13610776696011280 p^{6} T^{26} + 132447463647444 p^{8} T^{28} - 1138166350304 p^{10} T^{30} + 8334146485 p^{12} T^{32} - 49439326 p^{14} T^{34} + 221179 p^{16} T^{36} - 662 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( 1 - 772 T^{2} + 300974 T^{4} - 79250180 T^{6} + 15881528397 T^{8} - 2583290881840 T^{10} + 354721609143528 T^{12} - 42178231844928304 T^{14} + 4416281458244224626 T^{16} - \)\(41\!\cdots\!84\)\( T^{18} + \)\(34\!\cdots\!44\)\( T^{20} - \)\(41\!\cdots\!84\)\( p^{2} T^{22} + 4416281458244224626 p^{4} T^{24} - 42178231844928304 p^{6} T^{26} + 354721609143528 p^{8} T^{28} - 2583290881840 p^{10} T^{30} + 15881528397 p^{12} T^{32} - 79250180 p^{14} T^{34} + 300974 p^{16} T^{36} - 772 p^{18} T^{38} + p^{20} T^{40} \) | |
| 83 | \( ( 1 + 24 T + 894 T^{2} + 15600 T^{3} + 332865 T^{4} + 4597176 T^{5} + 71783808 T^{6} + 818073720 T^{7} + 10193779926 T^{8} + 97546659000 T^{9} + 1006811916964 T^{10} + 97546659000 p T^{11} + 10193779926 p^{2} T^{12} + 818073720 p^{3} T^{13} + 71783808 p^{4} T^{14} + 4597176 p^{5} T^{15} + 332865 p^{6} T^{16} + 15600 p^{7} T^{17} + 894 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 - 6 T + 186 T^{2} - 2824 T^{3} + 9733 T^{4} - 476660 T^{5} + 2104166 T^{6} - 32440014 T^{7} + 639120004 T^{8} - 1892373150 T^{9} + 80881921830 T^{10} - 470701967188 T^{11} + 5583842052083 T^{12} - 81722613898076 T^{13} + 320383685920070 T^{14} - 8669810929759270 T^{15} + 38773926433759431 T^{16} - 628346584296533016 T^{17} + 6870273527349716276 T^{18} - 34025577862123733188 T^{19} + \)\(80\!\cdots\!12\)\( T^{20} - 34025577862123733188 p T^{21} + 6870273527349716276 p^{2} T^{22} - 628346584296533016 p^{3} T^{23} + 38773926433759431 p^{4} T^{24} - 8669810929759270 p^{5} T^{25} + 320383685920070 p^{6} T^{26} - 81722613898076 p^{7} T^{27} + 5583842052083 p^{8} T^{28} - 470701967188 p^{9} T^{29} + 80881921830 p^{10} T^{30} - 1892373150 p^{11} T^{31} + 639120004 p^{12} T^{32} - 32440014 p^{13} T^{33} + 2104166 p^{14} T^{34} - 476660 p^{15} T^{35} + 9733 p^{16} T^{36} - 2824 p^{17} T^{37} + 186 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 66 T + 2594 T^{2} - 75372 T^{3} + 1779689 T^{4} - 35813304 T^{5} + 632655514 T^{6} - 10003781694 T^{7} + 143514046128 T^{8} - 1885840605930 T^{9} + 22842952299098 T^{10} - 255964711660032 T^{11} + 2654390580657267 T^{12} - 25383455174125188 T^{13} + 221670781665342902 T^{14} - 1730983594711449630 T^{15} + 11529148182391826187 T^{16} - 57231955975271800200 T^{17} + 81314086133490952836 T^{18} + \)\(24\!\cdots\!92\)\( T^{19} - \)\(35\!\cdots\!60\)\( T^{20} + \)\(24\!\cdots\!92\)\( p T^{21} + 81314086133490952836 p^{2} T^{22} - 57231955975271800200 p^{3} T^{23} + 11529148182391826187 p^{4} T^{24} - 1730983594711449630 p^{5} T^{25} + 221670781665342902 p^{6} T^{26} - 25383455174125188 p^{7} T^{27} + 2654390580657267 p^{8} T^{28} - 255964711660032 p^{9} T^{29} + 22842952299098 p^{10} T^{30} - 1885840605930 p^{11} T^{31} + 143514046128 p^{12} T^{32} - 10003781694 p^{13} T^{33} + 632655514 p^{14} T^{34} - 35813304 p^{15} T^{35} + 1779689 p^{16} T^{36} - 75372 p^{17} T^{37} + 2594 p^{18} T^{38} - 66 p^{19} T^{39} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.26889023441531115661583985077, −2.23418103201486542798163854047, −2.21403134661476178188291967679, −2.13622824670605872763107958002, −1.89512714422108636181565133897, −1.83253976670902487318384438658, −1.81124811755786337625005788185, −1.58526407203430459015873833937, −1.57582273888687760049923518933, −1.56914670491405406544045682754, −1.37713461139747705644991336144, −1.31197314872825566391898586363, −1.23736946958450352686670350882, −1.22455510612709162799641902281, −1.14389057678432726791314157561, −0.977426736453809618221803231864, −0.951536001703881398986142017293, −0.840239350458734116899931495616, −0.78202258286069750546902579239, −0.73880104053459999836669802110, −0.32614270682394806972251029428, −0.29623316250245102333529795452, −0.29177821090063338201057890855, −0.28939902591876292627098520729, −0.27557898599933907614561398826, 0.27557898599933907614561398826, 0.28939902591876292627098520729, 0.29177821090063338201057890855, 0.29623316250245102333529795452, 0.32614270682394806972251029428, 0.73880104053459999836669802110, 0.78202258286069750546902579239, 0.840239350458734116899931495616, 0.951536001703881398986142017293, 0.977426736453809618221803231864, 1.14389057678432726791314157561, 1.22455510612709162799641902281, 1.23736946958450352686670350882, 1.31197314872825566391898586363, 1.37713461139747705644991336144, 1.56914670491405406544045682754, 1.57582273888687760049923518933, 1.58526407203430459015873833937, 1.81124811755786337625005788185, 1.83253976670902487318384438658, 1.89512714422108636181565133897, 2.13622824670605872763107958002, 2.21403134661476178188291967679, 2.23418103201486542798163854047, 2.26889023441531115661583985077
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.