Dirichlet series
| L(s) = 1 | + 6·2-s + 9·3-s + 36·4-s + 24·5-s + 54·6-s − 63·7-s + 126·8-s + 72·9-s + 144·10-s + 111·11-s + 324·12-s − 18·13-s − 378·14-s + 216·15-s + 468·16-s − 546·17-s + 432·18-s + 90·19-s + 864·20-s − 567·21-s + 666·22-s + 312·23-s + 1.13e3·24-s + 711·25-s − 108·26-s + 549·27-s − 2.26e3·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 9/2·4-s + 2.14·5-s + 3.67·6-s − 3.40·7-s + 5.56·8-s + 8/3·9-s + 4.55·10-s + 3.04·11-s + 7.79·12-s − 0.384·13-s − 7.21·14-s + 3.71·15-s + 7.31·16-s − 7.78·17-s + 5.65·18-s + 1.08·19-s + 9.65·20-s − 5.89·21-s + 6.45·22-s + 2.82·23-s + 9.64·24-s + 5.68·25-s − 0.814·26-s + 3.91·27-s − 15.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{36} \cdot 7^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.83535\times 10^{10}\) |
| Root analytic conductor: | \(1.92798\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 3^{36} \cdot 7^{18} ,\ ( \ : [3/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(141.7977092\) |
| \(L(\frac12)\) | \(\approx\) | \(141.7977092\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - p^{2} T + p^{2} T^{2} + 2 p^{2} T^{3} + 47 p^{3} T^{4} - 101 p^{4} T^{5} - 164 p^{3} T^{6} + 149 p^{6} T^{7} + 7 p^{10} T^{8} - 766 p^{8} T^{9} + 7 p^{13} T^{10} + 149 p^{12} T^{11} - 164 p^{12} T^{12} - 101 p^{16} T^{13} + 47 p^{18} T^{14} + 2 p^{20} T^{15} + p^{23} T^{16} - p^{26} T^{17} + p^{27} T^{18} \) |
| 7 | \( ( 1 + p T + p^{2} T^{2} )^{9} \) | |
| good | 2 | \( 1 - 3 p T + 45 p T^{3} - 63 p^{2} T^{4} - 231 T^{5} + 2433 T^{6} - 1533 p T^{7} - 4923 T^{8} + 225 p^{6} T^{9} - 34659 T^{10} + 185313 T^{11} - 505137 T^{12} - 137019 p T^{13} + 1516167 p^{2} T^{14} - 1266237 p^{4} T^{15} + 1699731 p^{4} T^{16} + 753567 p^{7} T^{17} - 8774651 p^{6} T^{18} + 753567 p^{10} T^{19} + 1699731 p^{10} T^{20} - 1266237 p^{13} T^{21} + 1516167 p^{14} T^{22} - 137019 p^{16} T^{23} - 505137 p^{18} T^{24} + 185313 p^{21} T^{25} - 34659 p^{24} T^{26} + 225 p^{33} T^{27} - 4923 p^{30} T^{28} - 1533 p^{34} T^{29} + 2433 p^{36} T^{30} - 231 p^{39} T^{31} - 63 p^{44} T^{32} + 45 p^{46} T^{33} - 3 p^{52} T^{35} + p^{54} T^{36} \) |
| 5 | \( 1 - 24 T - 27 p T^{2} + 4284 T^{3} + 54072 T^{4} - 313044 T^{5} - 15740079 T^{6} - 11155656 p T^{7} + 2773664253 T^{8} + 25738770672 T^{9} - 265166877816 T^{10} - 1180059725316 p T^{11} + 589682928372 T^{12} + 893960315200164 T^{13} + 5517977552150157 T^{14} - 91672964034914088 T^{15} - 1281826000386231867 T^{16} + 4703053884869404056 T^{17} + \)\(18\!\cdots\!24\)\( T^{18} + 4703053884869404056 p^{3} T^{19} - 1281826000386231867 p^{6} T^{20} - 91672964034914088 p^{9} T^{21} + 5517977552150157 p^{12} T^{22} + 893960315200164 p^{15} T^{23} + 589682928372 p^{18} T^{24} - 1180059725316 p^{22} T^{25} - 265166877816 p^{24} T^{26} + 25738770672 p^{27} T^{27} + 2773664253 p^{30} T^{28} - 11155656 p^{34} T^{29} - 15740079 p^{36} T^{30} - 313044 p^{39} T^{31} + 54072 p^{42} T^{32} + 4284 p^{45} T^{33} - 27 p^{49} T^{34} - 24 p^{51} T^{35} + p^{54} T^{36} \) | |
| 11 | \( 1 - 111 T + 702 T^{2} + 275031 T^{3} + 1265238 T^{4} - 928261857 T^{5} + 11439493518 T^{6} + 1092829962687 T^{7} + 3152075996304 T^{8} - 2138272282554579 T^{9} + 23019715641124836 T^{10} + 882464328696498459 T^{11} + 22286657835297708522 T^{12} - \)\(24\!\cdots\!53\)\( T^{13} + \)\(90\!\cdots\!78\)\( T^{14} - \)\(89\!\cdots\!41\)\( T^{15} - \)\(63\!\cdots\!02\)\( T^{16} - \)\(27\!\cdots\!59\)\( T^{17} + \)\(30\!\cdots\!06\)\( T^{18} - \)\(27\!\cdots\!59\)\( p^{3} T^{19} - \)\(63\!\cdots\!02\)\( p^{6} T^{20} - \)\(89\!\cdots\!41\)\( p^{9} T^{21} + \)\(90\!\cdots\!78\)\( p^{12} T^{22} - \)\(24\!\cdots\!53\)\( p^{15} T^{23} + 22286657835297708522 p^{18} T^{24} + 882464328696498459 p^{21} T^{25} + 23019715641124836 p^{24} T^{26} - 2138272282554579 p^{27} T^{27} + 3152075996304 p^{30} T^{28} + 1092829962687 p^{33} T^{29} + 11439493518 p^{36} T^{30} - 928261857 p^{39} T^{31} + 1265238 p^{42} T^{32} + 275031 p^{45} T^{33} + 702 p^{48} T^{34} - 111 p^{51} T^{35} + p^{54} T^{36} \) | |
| 13 | \( 1 + 18 T - 11790 T^{2} - 25680 p T^{3} + 67296114 T^{4} + 2377926054 T^{5} - 19094308710 p T^{6} - 9077200849170 T^{7} + 55680918814026 p T^{8} + 20640745941737936 T^{9} - 159999032891841750 p T^{10} - 30051189439357231350 T^{11} + \)\(61\!\cdots\!35\)\( T^{12} + \)\(39\!\cdots\!04\)\( T^{13} - \)\(16\!\cdots\!12\)\( T^{14} - \)\(74\!\cdots\!72\)\( T^{15} + \)\(38\!\cdots\!44\)\( T^{16} + \)\(80\!\cdots\!80\)\( T^{17} - \)\(84\!\cdots\!16\)\( T^{18} + \)\(80\!\cdots\!80\)\( p^{3} T^{19} + \)\(38\!\cdots\!44\)\( p^{6} T^{20} - \)\(74\!\cdots\!72\)\( p^{9} T^{21} - \)\(16\!\cdots\!12\)\( p^{12} T^{22} + \)\(39\!\cdots\!04\)\( p^{15} T^{23} + \)\(61\!\cdots\!35\)\( p^{18} T^{24} - 30051189439357231350 p^{21} T^{25} - 159999032891841750 p^{25} T^{26} + 20640745941737936 p^{27} T^{27} + 55680918814026 p^{31} T^{28} - 9077200849170 p^{33} T^{29} - 19094308710 p^{37} T^{30} + 2377926054 p^{39} T^{31} + 67296114 p^{42} T^{32} - 25680 p^{46} T^{33} - 11790 p^{48} T^{34} + 18 p^{51} T^{35} + p^{54} T^{36} \) | |
| 17 | \( ( 1 + 273 T + 54171 T^{2} + 7772238 T^{3} + 968957271 T^{4} + 102002032857 T^{5} + 9773002638042 T^{6} + 833158680136113 T^{7} + 66278147938872399 T^{8} + 4792749919630444854 T^{9} + 66278147938872399 p^{3} T^{10} + 833158680136113 p^{6} T^{11} + 9773002638042 p^{9} T^{12} + 102002032857 p^{12} T^{13} + 968957271 p^{15} T^{14} + 7772238 p^{18} T^{15} + 54171 p^{21} T^{16} + 273 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 19 | \( ( 1 - 45 T + 38925 T^{2} - 2601348 T^{3} + 709193457 T^{4} - 62259072669 T^{5} + 8287221918066 T^{6} - 840378579772851 T^{7} + 71702530972816086 T^{8} - 7149895439436435791 T^{9} + 71702530972816086 p^{3} T^{10} - 840378579772851 p^{6} T^{11} + 8287221918066 p^{9} T^{12} - 62259072669 p^{12} T^{13} + 709193457 p^{15} T^{14} - 2601348 p^{18} T^{15} + 38925 p^{21} T^{16} - 45 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 23 | \( 1 - 312 T + 23652 T^{2} + 330156 p T^{3} - 2146907142 T^{4} + 183846066258 T^{5} + 18462658479600 T^{6} - 6811755519336540 T^{7} + 751342821079870584 T^{8} + 1069713603046935900 T^{9} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!60\)\( T^{11} - \)\(10\!\cdots\!72\)\( T^{12} - \)\(11\!\cdots\!42\)\( T^{13} + \)\(31\!\cdots\!38\)\( T^{14} - \)\(31\!\cdots\!84\)\( T^{15} + \)\(51\!\cdots\!76\)\( T^{16} + \)\(31\!\cdots\!32\)\( T^{17} - \)\(52\!\cdots\!70\)\( T^{18} + \)\(31\!\cdots\!32\)\( p^{3} T^{19} + \)\(51\!\cdots\!76\)\( p^{6} T^{20} - \)\(31\!\cdots\!84\)\( p^{9} T^{21} + \)\(31\!\cdots\!38\)\( p^{12} T^{22} - \)\(11\!\cdots\!42\)\( p^{15} T^{23} - \)\(10\!\cdots\!72\)\( p^{18} T^{24} + \)\(19\!\cdots\!60\)\( p^{21} T^{25} - \)\(12\!\cdots\!80\)\( p^{24} T^{26} + 1069713603046935900 p^{27} T^{27} + 751342821079870584 p^{30} T^{28} - 6811755519336540 p^{33} T^{29} + 18462658479600 p^{36} T^{30} + 183846066258 p^{39} T^{31} - 2146907142 p^{42} T^{32} + 330156 p^{46} T^{33} + 23652 p^{48} T^{34} - 312 p^{51} T^{35} + p^{54} T^{36} \) | |
| 29 | \( 1 - 378 T - 60390 T^{2} + 32343264 T^{3} + 3419158491 T^{4} - 1712659382352 T^{5} - 178308643692966 T^{6} + 67132136829713688 T^{7} + 258640808694435585 p T^{8} - \)\(19\!\cdots\!76\)\( T^{9} - \)\(27\!\cdots\!36\)\( T^{10} + \)\(15\!\cdots\!02\)\( p T^{11} + \)\(86\!\cdots\!44\)\( T^{12} - \)\(83\!\cdots\!12\)\( T^{13} - \)\(23\!\cdots\!01\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(62\!\cdots\!42\)\( T^{16} - \)\(79\!\cdots\!32\)\( T^{17} - \)\(15\!\cdots\!07\)\( T^{18} - \)\(79\!\cdots\!32\)\( p^{3} T^{19} + \)\(62\!\cdots\!42\)\( p^{6} T^{20} + \)\(11\!\cdots\!00\)\( p^{9} T^{21} - \)\(23\!\cdots\!01\)\( p^{12} T^{22} - \)\(83\!\cdots\!12\)\( p^{15} T^{23} + \)\(86\!\cdots\!44\)\( p^{18} T^{24} + \)\(15\!\cdots\!02\)\( p^{22} T^{25} - \)\(27\!\cdots\!36\)\( p^{24} T^{26} - \)\(19\!\cdots\!76\)\( p^{27} T^{27} + 258640808694435585 p^{31} T^{28} + 67132136829713688 p^{33} T^{29} - 178308643692966 p^{36} T^{30} - 1712659382352 p^{39} T^{31} + 3419158491 p^{42} T^{32} + 32343264 p^{45} T^{33} - 60390 p^{48} T^{34} - 378 p^{51} T^{35} + p^{54} T^{36} \) | |
| 31 | \( 1 + 18 T - 151704 T^{2} - 4525968 T^{3} + 10857610485 T^{4} + 400645742382 T^{5} - 526464550327740 T^{6} - 16887794293815498 T^{7} + 21797169327038746179 T^{8} + \)\(35\!\cdots\!70\)\( T^{9} - \)\(87\!\cdots\!96\)\( T^{10} - \)\(45\!\cdots\!86\)\( T^{11} + \)\(33\!\cdots\!94\)\( T^{12} + \)\(16\!\cdots\!14\)\( T^{13} - \)\(11\!\cdots\!61\)\( T^{14} - \)\(77\!\cdots\!40\)\( T^{15} + \)\(36\!\cdots\!74\)\( T^{16} + \)\(12\!\cdots\!18\)\( T^{17} - \)\(10\!\cdots\!65\)\( T^{18} + \)\(12\!\cdots\!18\)\( p^{3} T^{19} + \)\(36\!\cdots\!74\)\( p^{6} T^{20} - \)\(77\!\cdots\!40\)\( p^{9} T^{21} - \)\(11\!\cdots\!61\)\( p^{12} T^{22} + \)\(16\!\cdots\!14\)\( p^{15} T^{23} + \)\(33\!\cdots\!94\)\( p^{18} T^{24} - \)\(45\!\cdots\!86\)\( p^{21} T^{25} - \)\(87\!\cdots\!96\)\( p^{24} T^{26} + \)\(35\!\cdots\!70\)\( p^{27} T^{27} + 21797169327038746179 p^{30} T^{28} - 16887794293815498 p^{33} T^{29} - 526464550327740 p^{36} T^{30} + 400645742382 p^{39} T^{31} + 10857610485 p^{42} T^{32} - 4525968 p^{45} T^{33} - 151704 p^{48} T^{34} + 18 p^{51} T^{35} + p^{54} T^{36} \) | |
| 37 | \( ( 1 + 36 T + 281574 T^{2} + 8355144 T^{3} + 38702854413 T^{4} + 1148561406576 T^{5} + 3497796368226528 T^{6} + 108726691647213162 T^{7} + \)\(23\!\cdots\!32\)\( T^{8} + \)\(68\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!32\)\( p^{3} T^{10} + 108726691647213162 p^{6} T^{11} + 3497796368226528 p^{9} T^{12} + 1148561406576 p^{12} T^{13} + 38702854413 p^{15} T^{14} + 8355144 p^{18} T^{15} + 281574 p^{21} T^{16} + 36 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 41 | \( 1 - 477 T - 127026 T^{2} + 100179549 T^{3} + 2571445512 T^{4} - 11016504815049 T^{5} + 1080814638078678 T^{6} + 755078533104102165 T^{7} - \)\(14\!\cdots\!48\)\( T^{8} - \)\(45\!\cdots\!21\)\( T^{9} + \)\(14\!\cdots\!62\)\( T^{10} + \)\(27\!\cdots\!73\)\( T^{11} - \)\(13\!\cdots\!36\)\( T^{12} - \)\(11\!\cdots\!33\)\( T^{13} + \)\(10\!\cdots\!90\)\( T^{14} + \)\(91\!\cdots\!65\)\( T^{15} - \)\(58\!\cdots\!00\)\( T^{16} + \)\(79\!\cdots\!71\)\( T^{17} + \)\(33\!\cdots\!44\)\( T^{18} + \)\(79\!\cdots\!71\)\( p^{3} T^{19} - \)\(58\!\cdots\!00\)\( p^{6} T^{20} + \)\(91\!\cdots\!65\)\( p^{9} T^{21} + \)\(10\!\cdots\!90\)\( p^{12} T^{22} - \)\(11\!\cdots\!33\)\( p^{15} T^{23} - \)\(13\!\cdots\!36\)\( p^{18} T^{24} + \)\(27\!\cdots\!73\)\( p^{21} T^{25} + \)\(14\!\cdots\!62\)\( p^{24} T^{26} - \)\(45\!\cdots\!21\)\( p^{27} T^{27} - \)\(14\!\cdots\!48\)\( p^{30} T^{28} + 755078533104102165 p^{33} T^{29} + 1080814638078678 p^{36} T^{30} - 11016504815049 p^{39} T^{31} + 2571445512 p^{42} T^{32} + 100179549 p^{45} T^{33} - 127026 p^{48} T^{34} - 477 p^{51} T^{35} + p^{54} T^{36} \) | |
| 43 | \( 1 - 171 T - 426483 T^{2} + 41179794 T^{3} + 94371482016 T^{4} - 2792580312384 T^{5} - 14659227739710609 T^{6} - 180703342390066785 T^{7} + \)\(18\!\cdots\!01\)\( T^{8} + \)\(46\!\cdots\!68\)\( T^{9} - \)\(19\!\cdots\!62\)\( T^{10} - \)\(38\!\cdots\!24\)\( T^{11} + \)\(20\!\cdots\!12\)\( T^{12} + \)\(14\!\cdots\!10\)\( T^{13} - \)\(19\!\cdots\!79\)\( T^{14} + \)\(18\!\cdots\!13\)\( T^{15} + \)\(17\!\cdots\!51\)\( T^{16} - \)\(25\!\cdots\!86\)\( T^{17} - \)\(14\!\cdots\!28\)\( T^{18} - \)\(25\!\cdots\!86\)\( p^{3} T^{19} + \)\(17\!\cdots\!51\)\( p^{6} T^{20} + \)\(18\!\cdots\!13\)\( p^{9} T^{21} - \)\(19\!\cdots\!79\)\( p^{12} T^{22} + \)\(14\!\cdots\!10\)\( p^{15} T^{23} + \)\(20\!\cdots\!12\)\( p^{18} T^{24} - \)\(38\!\cdots\!24\)\( p^{21} T^{25} - \)\(19\!\cdots\!62\)\( p^{24} T^{26} + \)\(46\!\cdots\!68\)\( p^{27} T^{27} + \)\(18\!\cdots\!01\)\( p^{30} T^{28} - 180703342390066785 p^{33} T^{29} - 14659227739710609 p^{36} T^{30} - 2792580312384 p^{39} T^{31} + 94371482016 p^{42} T^{32} + 41179794 p^{45} T^{33} - 426483 p^{48} T^{34} - 171 p^{51} T^{35} + p^{54} T^{36} \) | |
| 47 | \( 1 - 654 T - 251046 T^{2} + 153655848 T^{3} + 85771776033 T^{4} - 28676474262762 T^{5} - 16055837602998162 T^{6} + 2281442852927326800 T^{7} + \)\(22\!\cdots\!57\)\( T^{8} + \)\(20\!\cdots\!18\)\( T^{9} - \)\(18\!\cdots\!38\)\( T^{10} - \)\(31\!\cdots\!46\)\( T^{11} + \)\(24\!\cdots\!44\)\( T^{12} + \)\(48\!\cdots\!18\)\( T^{13} + \)\(18\!\cdots\!23\)\( T^{14} - \)\(35\!\cdots\!40\)\( T^{15} - \)\(36\!\cdots\!04\)\( T^{16} + \)\(17\!\cdots\!68\)\( T^{17} + \)\(42\!\cdots\!95\)\( T^{18} + \)\(17\!\cdots\!68\)\( p^{3} T^{19} - \)\(36\!\cdots\!04\)\( p^{6} T^{20} - \)\(35\!\cdots\!40\)\( p^{9} T^{21} + \)\(18\!\cdots\!23\)\( p^{12} T^{22} + \)\(48\!\cdots\!18\)\( p^{15} T^{23} + \)\(24\!\cdots\!44\)\( p^{18} T^{24} - \)\(31\!\cdots\!46\)\( p^{21} T^{25} - \)\(18\!\cdots\!38\)\( p^{24} T^{26} + \)\(20\!\cdots\!18\)\( p^{27} T^{27} + \)\(22\!\cdots\!57\)\( p^{30} T^{28} + 2281442852927326800 p^{33} T^{29} - 16055837602998162 p^{36} T^{30} - 28676474262762 p^{39} T^{31} + 85771776033 p^{42} T^{32} + 153655848 p^{45} T^{33} - 251046 p^{48} T^{34} - 654 p^{51} T^{35} + p^{54} T^{36} \) | |
| 53 | \( ( 1 + 948 T + 1389591 T^{2} + 911764422 T^{3} + 747220561587 T^{4} + 372226870112574 T^{5} + 221418756940245828 T^{6} + 89663338994683246554 T^{7} + \)\(43\!\cdots\!01\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(43\!\cdots\!01\)\( p^{3} T^{10} + 89663338994683246554 p^{6} T^{11} + 221418756940245828 p^{9} T^{12} + 372226870112574 p^{12} T^{13} + 747220561587 p^{15} T^{14} + 911764422 p^{18} T^{15} + 1389591 p^{21} T^{16} + 948 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 59 | \( 1 - 957 T - 98361 T^{2} + 735477084 T^{3} - 430472366334 T^{4} - 49716354336762 T^{5} + 192187410340337805 T^{6} - 91904385221774360049 T^{7} - \)\(61\!\cdots\!81\)\( T^{8} + \)\(28\!\cdots\!02\)\( T^{9} - \)\(12\!\cdots\!44\)\( T^{10} - \)\(36\!\cdots\!02\)\( T^{11} + \)\(30\!\cdots\!30\)\( T^{12} - \)\(12\!\cdots\!04\)\( T^{13} - \)\(90\!\cdots\!67\)\( T^{14} + \)\(83\!\cdots\!23\)\( p^{2} T^{15} - \)\(91\!\cdots\!89\)\( T^{16} - \)\(23\!\cdots\!18\)\( T^{17} + \)\(26\!\cdots\!32\)\( T^{18} - \)\(23\!\cdots\!18\)\( p^{3} T^{19} - \)\(91\!\cdots\!89\)\( p^{6} T^{20} + \)\(83\!\cdots\!23\)\( p^{11} T^{21} - \)\(90\!\cdots\!67\)\( p^{12} T^{22} - \)\(12\!\cdots\!04\)\( p^{15} T^{23} + \)\(30\!\cdots\!30\)\( p^{18} T^{24} - \)\(36\!\cdots\!02\)\( p^{21} T^{25} - \)\(12\!\cdots\!44\)\( p^{24} T^{26} + \)\(28\!\cdots\!02\)\( p^{27} T^{27} - \)\(61\!\cdots\!81\)\( p^{30} T^{28} - 91904385221774360049 p^{33} T^{29} + 192187410340337805 p^{36} T^{30} - 49716354336762 p^{39} T^{31} - 430472366334 p^{42} T^{32} + 735477084 p^{45} T^{33} - 98361 p^{48} T^{34} - 957 p^{51} T^{35} + p^{54} T^{36} \) | |
| 61 | \( 1 - 198 T - 887778 T^{2} - 287437260 T^{3} + 528889911822 T^{4} + 307113774288822 T^{5} - 116372190688624860 T^{6} - \)\(17\!\cdots\!38\)\( T^{7} - \)\(13\!\cdots\!26\)\( T^{8} + \)\(50\!\cdots\!72\)\( T^{9} + \)\(22\!\cdots\!86\)\( T^{10} - \)\(68\!\cdots\!58\)\( T^{11} - \)\(76\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!82\)\( T^{13} + \)\(12\!\cdots\!98\)\( T^{14} + \)\(60\!\cdots\!76\)\( T^{15} + \)\(33\!\cdots\!70\)\( T^{16} - \)\(77\!\cdots\!54\)\( T^{17} - \)\(41\!\cdots\!02\)\( T^{18} - \)\(77\!\cdots\!54\)\( p^{3} T^{19} + \)\(33\!\cdots\!70\)\( p^{6} T^{20} + \)\(60\!\cdots\!76\)\( p^{9} T^{21} + \)\(12\!\cdots\!98\)\( p^{12} T^{22} - \)\(11\!\cdots\!82\)\( p^{15} T^{23} - \)\(76\!\cdots\!20\)\( p^{18} T^{24} - \)\(68\!\cdots\!58\)\( p^{21} T^{25} + \)\(22\!\cdots\!86\)\( p^{24} T^{26} + \)\(50\!\cdots\!72\)\( p^{27} T^{27} - \)\(13\!\cdots\!26\)\( p^{30} T^{28} - \)\(17\!\cdots\!38\)\( p^{33} T^{29} - 116372190688624860 p^{36} T^{30} + 307113774288822 p^{39} T^{31} + 528889911822 p^{42} T^{32} - 287437260 p^{45} T^{33} - 887778 p^{48} T^{34} - 198 p^{51} T^{35} + p^{54} T^{36} \) | |
| 67 | \( 1 - 333 T - 1732230 T^{2} + 528423009 T^{3} + 1536024854412 T^{4} - 412920519370443 T^{5} - 968545743628278552 T^{6} + \)\(22\!\cdots\!51\)\( T^{7} + \)\(49\!\cdots\!18\)\( T^{8} - \)\(90\!\cdots\!95\)\( T^{9} - \)\(21\!\cdots\!74\)\( T^{10} + \)\(29\!\cdots\!79\)\( T^{11} + \)\(85\!\cdots\!76\)\( T^{12} - \)\(74\!\cdots\!11\)\( T^{13} - \)\(30\!\cdots\!04\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{15} + \)\(10\!\cdots\!50\)\( T^{16} - \)\(15\!\cdots\!09\)\( T^{17} - \)\(31\!\cdots\!66\)\( T^{18} - \)\(15\!\cdots\!09\)\( p^{3} T^{19} + \)\(10\!\cdots\!50\)\( p^{6} T^{20} + \)\(14\!\cdots\!01\)\( p^{9} T^{21} - \)\(30\!\cdots\!04\)\( p^{12} T^{22} - \)\(74\!\cdots\!11\)\( p^{15} T^{23} + \)\(85\!\cdots\!76\)\( p^{18} T^{24} + \)\(29\!\cdots\!79\)\( p^{21} T^{25} - \)\(21\!\cdots\!74\)\( p^{24} T^{26} - \)\(90\!\cdots\!95\)\( p^{27} T^{27} + \)\(49\!\cdots\!18\)\( p^{30} T^{28} + \)\(22\!\cdots\!51\)\( p^{33} T^{29} - 968545743628278552 p^{36} T^{30} - 412920519370443 p^{39} T^{31} + 1536024854412 p^{42} T^{32} + 528423009 p^{45} T^{33} - 1732230 p^{48} T^{34} - 333 p^{51} T^{35} + p^{54} T^{36} \) | |
| 71 | \( ( 1 + 2826 T + 5263047 T^{2} + 6924084732 T^{3} + 7456807599435 T^{4} + 6647912633823210 T^{5} + 5231634779026552794 T^{6} + \)\(36\!\cdots\!26\)\( T^{7} + \)\(23\!\cdots\!18\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!18\)\( p^{3} T^{10} + \)\(36\!\cdots\!26\)\( p^{6} T^{11} + 5231634779026552794 p^{9} T^{12} + 6647912633823210 p^{12} T^{13} + 7456807599435 p^{15} T^{14} + 6924084732 p^{18} T^{15} + 5263047 p^{21} T^{16} + 2826 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 73 | \( ( 1 - 153 T + 2081772 T^{2} - 57071499 T^{3} + 1958269856463 T^{4} + 263808176386464 T^{5} + 1132889252675869788 T^{6} + \)\(34\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!36\)\( T^{8} + \)\(18\!\cdots\!04\)\( T^{9} + \)\(49\!\cdots\!36\)\( p^{3} T^{10} + \)\(34\!\cdots\!20\)\( p^{6} T^{11} + 1132889252675869788 p^{9} T^{12} + 263808176386464 p^{12} T^{13} + 1958269856463 p^{15} T^{14} - 57071499 p^{18} T^{15} + 2081772 p^{21} T^{16} - 153 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 79 | \( 1 + 1152 T - 2310273 T^{2} - 2463307860 T^{3} + 3542715673650 T^{4} + 3068374894454970 T^{5} - 3930405513412520391 T^{6} - \)\(26\!\cdots\!06\)\( T^{7} + \)\(33\!\cdots\!23\)\( T^{8} + \)\(17\!\cdots\!64\)\( T^{9} - \)\(23\!\cdots\!14\)\( T^{10} - \)\(11\!\cdots\!42\)\( p T^{11} + \)\(13\!\cdots\!36\)\( T^{12} + \)\(37\!\cdots\!18\)\( T^{13} - \)\(63\!\cdots\!91\)\( T^{14} - \)\(12\!\cdots\!96\)\( T^{15} + \)\(28\!\cdots\!25\)\( T^{16} + \)\(20\!\cdots\!76\)\( T^{17} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(20\!\cdots\!76\)\( p^{3} T^{19} + \)\(28\!\cdots\!25\)\( p^{6} T^{20} - \)\(12\!\cdots\!96\)\( p^{9} T^{21} - \)\(63\!\cdots\!91\)\( p^{12} T^{22} + \)\(37\!\cdots\!18\)\( p^{15} T^{23} + \)\(13\!\cdots\!36\)\( p^{18} T^{24} - \)\(11\!\cdots\!42\)\( p^{22} T^{25} - \)\(23\!\cdots\!14\)\( p^{24} T^{26} + \)\(17\!\cdots\!64\)\( p^{27} T^{27} + \)\(33\!\cdots\!23\)\( p^{30} T^{28} - \)\(26\!\cdots\!06\)\( p^{33} T^{29} - 3930405513412520391 p^{36} T^{30} + 3068374894454970 p^{39} T^{31} + 3542715673650 p^{42} T^{32} - 2463307860 p^{45} T^{33} - 2310273 p^{48} T^{34} + 1152 p^{51} T^{35} + p^{54} T^{36} \) | |
| 83 | \( 1 - 1890 T - 1259001 T^{2} + 3565334466 T^{3} + 1727740592604 T^{4} - 4244684325885126 T^{5} - 2320472508559383597 T^{6} + \)\(38\!\cdots\!36\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} - \)\(26\!\cdots\!76\)\( T^{9} - \)\(20\!\cdots\!19\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!84\)\( T^{12} - \)\(64\!\cdots\!34\)\( T^{13} - \)\(90\!\cdots\!89\)\( T^{14} + \)\(22\!\cdots\!86\)\( T^{15} + \)\(51\!\cdots\!52\)\( T^{16} - \)\(46\!\cdots\!14\)\( T^{17} - \)\(28\!\cdots\!07\)\( T^{18} - \)\(46\!\cdots\!14\)\( p^{3} T^{19} + \)\(51\!\cdots\!52\)\( p^{6} T^{20} + \)\(22\!\cdots\!86\)\( p^{9} T^{21} - \)\(90\!\cdots\!89\)\( p^{12} T^{22} - \)\(64\!\cdots\!34\)\( p^{15} T^{23} + \)\(14\!\cdots\!84\)\( p^{18} T^{24} + \)\(14\!\cdots\!40\)\( p^{21} T^{25} - \)\(20\!\cdots\!19\)\( p^{24} T^{26} - \)\(26\!\cdots\!76\)\( p^{27} T^{27} + \)\(24\!\cdots\!66\)\( p^{30} T^{28} + \)\(38\!\cdots\!36\)\( p^{33} T^{29} - 2320472508559383597 p^{36} T^{30} - 4244684325885126 p^{39} T^{31} + 1727740592604 p^{42} T^{32} + 3565334466 p^{45} T^{33} - 1259001 p^{48} T^{34} - 1890 p^{51} T^{35} + p^{54} T^{36} \) | |
| 89 | \( ( 1 + 1302 T + 2637747 T^{2} + 1784744028 T^{3} + 3054189577995 T^{4} + 2033343768333426 T^{5} + 3526920405043686474 T^{6} + \)\(20\!\cdots\!02\)\( T^{7} + \)\(27\!\cdots\!55\)\( T^{8} + \)\(12\!\cdots\!16\)\( T^{9} + \)\(27\!\cdots\!55\)\( p^{3} T^{10} + \)\(20\!\cdots\!02\)\( p^{6} T^{11} + 3526920405043686474 p^{9} T^{12} + 2033343768333426 p^{12} T^{13} + 3054189577995 p^{15} T^{14} + 1784744028 p^{18} T^{15} + 2637747 p^{21} T^{16} + 1302 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 97 | \( 1 - 1737 T - 1233891 T^{2} + 3264015786 T^{3} + 895203537786 T^{4} - 2970294593054454 T^{5} - 210857292003178851 T^{6} + \)\(61\!\cdots\!43\)\( T^{7} - \)\(50\!\cdots\!07\)\( T^{8} + \)\(33\!\cdots\!80\)\( T^{9} - \)\(11\!\cdots\!72\)\( T^{10} - \)\(40\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!68\)\( T^{12} + \)\(30\!\cdots\!96\)\( T^{13} - \)\(36\!\cdots\!15\)\( T^{14} - \)\(14\!\cdots\!61\)\( T^{15} - \)\(16\!\cdots\!15\)\( T^{16} - \)\(29\!\cdots\!38\)\( T^{17} + \)\(32\!\cdots\!78\)\( T^{18} - \)\(29\!\cdots\!38\)\( p^{3} T^{19} - \)\(16\!\cdots\!15\)\( p^{6} T^{20} - \)\(14\!\cdots\!61\)\( p^{9} T^{21} - \)\(36\!\cdots\!15\)\( p^{12} T^{22} + \)\(30\!\cdots\!96\)\( p^{15} T^{23} + \)\(15\!\cdots\!68\)\( p^{18} T^{24} - \)\(40\!\cdots\!96\)\( p^{21} T^{25} - \)\(11\!\cdots\!72\)\( p^{24} T^{26} + \)\(33\!\cdots\!80\)\( p^{27} T^{27} - \)\(50\!\cdots\!07\)\( p^{30} T^{28} + \)\(61\!\cdots\!43\)\( p^{33} T^{29} - 210857292003178851 p^{36} T^{30} - 2970294593054454 p^{39} T^{31} + 895203537786 p^{42} T^{32} + 3264015786 p^{45} T^{33} - 1233891 p^{48} T^{34} - 1737 p^{51} T^{35} + p^{54} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.83091700397251616567065446812, −3.65289539877669065948764830495, −3.45314056085812467074564592341, −3.32437335684068180092811227732, −3.11939027077760415601570324107, −3.10216660614921116089482640976, −3.03508732191702789188761122589, −3.02960595810379522019301643377, −2.96246029510722675054489363908, −2.94827469417589733591298550387, −2.91635161591254694752848750279, −2.54302131839918710492957752909, −2.47962668650558750285892523886, −2.34525121822386865237573830169, −2.30332381371017785866442933859, −2.07661515656162415773604163482, −1.90400248767647000556374290544, −1.89147752193614575608030314772, −1.70753783747005817348785766960, −1.37792147784222754215844223759, −1.12934365307258046161278215150, −1.11941459183221144933306558225, −0.962290245286800925704588716498, −0.55701310006217309717117682467, −0.23998306708814417874906059561, 0.23998306708814417874906059561, 0.55701310006217309717117682467, 0.962290245286800925704588716498, 1.11941459183221144933306558225, 1.12934365307258046161278215150, 1.37792147784222754215844223759, 1.70753783747005817348785766960, 1.89147752193614575608030314772, 1.90400248767647000556374290544, 2.07661515656162415773604163482, 2.30332381371017785866442933859, 2.34525121822386865237573830169, 2.47962668650558750285892523886, 2.54302131839918710492957752909, 2.91635161591254694752848750279, 2.94827469417589733591298550387, 2.96246029510722675054489363908, 3.02960595810379522019301643377, 3.03508732191702789188761122589, 3.10216660614921116089482640976, 3.11939027077760415601570324107, 3.32437335684068180092811227732, 3.45314056085812467074564592341, 3.65289539877669065948764830495, 3.83091700397251616567065446812
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.