Dirichlet series
| L(s) = 1 | − 2·3-s + 16·4-s + 30·5-s + 150·7-s + 219·9-s − 30·11-s − 32·12-s − 510·13-s − 60·15-s + 64·16-s + 1.77e3·17-s + 1.02e3·19-s + 480·20-s − 300·21-s − 2.42e3·23-s + 1.27e3·25-s + 900·27-s + 2.40e3·28-s + 4.89e3·29-s + 602·31-s + 60·33-s + 4.50e3·35-s + 3.50e3·36-s − 4.51e3·37-s + 1.02e3·39-s + 1.29e3·41-s − 480·44-s + ⋯ |
| L(s) = 1 | − 2/9·3-s + 4-s + 6/5·5-s + 3.06·7-s + 2.70·9-s − 0.247·11-s − 2/9·12-s − 3.01·13-s − 0.266·15-s + 1/4·16-s + 6.12·17-s + 2.82·19-s + 6/5·20-s − 0.680·21-s − 4.58·23-s + 2.03·25-s + 1.23·27-s + 3.06·28-s + 5.81·29-s + 0.626·31-s + 0.0550·33-s + 3.67·35-s + 2.70·36-s − 3.30·37-s + 0.670·39-s + 0.767·41-s − 0.247·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(511767.\) |
| Root analytic conductor: | \(1.50802\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 11^{16} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(37.00618962\) |
| \(L(\frac12)\) | \(\approx\) | \(37.00618962\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( 1 + 30 T + 1681 p T^{2} + 1080 p^{2} T^{3} + 326245 p^{3} T^{4} + 712380 p^{4} T^{5} + 46974899 p^{5} T^{6} + 8404830 p^{6} T^{7} + 607189384 p^{8} T^{8} + 8404830 p^{10} T^{9} + 46974899 p^{13} T^{10} + 712380 p^{16} T^{11} + 326245 p^{19} T^{12} + 1080 p^{22} T^{13} + 1681 p^{25} T^{14} + 30 p^{28} T^{15} + p^{32} T^{16} \) | |
| good | 3 | \( 1 + 2 T - 215 T^{2} - 1768 T^{3} + 2776 p^{2} T^{4} + 460318 T^{5} - 355948 p T^{6} - 60307666 T^{7} - 195306248 T^{8} + 1754921896 p T^{9} + 4876523695 p^{2} T^{10} - 10949202854 p^{3} T^{11} - 18570439531 p^{5} T^{12} + 36452194768 p^{5} T^{13} + 160941508432 p^{7} T^{14} - 22126948900 p^{7} T^{15} - 4060841715512 p^{8} T^{16} - 22126948900 p^{11} T^{17} + 160941508432 p^{15} T^{18} + 36452194768 p^{17} T^{19} - 18570439531 p^{21} T^{20} - 10949202854 p^{23} T^{21} + 4876523695 p^{26} T^{22} + 1754921896 p^{29} T^{23} - 195306248 p^{32} T^{24} - 60307666 p^{36} T^{25} - 355948 p^{41} T^{26} + 460318 p^{44} T^{27} + 2776 p^{50} T^{28} - 1768 p^{52} T^{29} - 215 p^{56} T^{30} + 2 p^{60} T^{31} + p^{64} T^{32} \) |
| 5 | \( 1 - 6 p T - 371 T^{2} + 216 p^{3} T^{3} - 800188 T^{4} + 2758158 p T^{5} + 471406872 T^{6} - 5715708978 p T^{7} + 21089345396 p^{2} T^{8} + 477286316664 p T^{9} - 345700002004657 T^{10} + 65055865322142 p^{3} T^{11} - 92727527079481653 T^{12} - 298818232874347824 p T^{13} + \)\(10\!\cdots\!16\)\( T^{14} - \)\(21\!\cdots\!64\)\( p T^{15} - \)\(15\!\cdots\!04\)\( T^{16} - \)\(21\!\cdots\!64\)\( p^{5} T^{17} + \)\(10\!\cdots\!16\)\( p^{8} T^{18} - 298818232874347824 p^{13} T^{19} - 92727527079481653 p^{16} T^{20} + 65055865322142 p^{23} T^{21} - 345700002004657 p^{24} T^{22} + 477286316664 p^{29} T^{23} + 21089345396 p^{34} T^{24} - 5715708978 p^{37} T^{25} + 471406872 p^{40} T^{26} + 2758158 p^{45} T^{27} - 800188 p^{48} T^{28} + 216 p^{55} T^{29} - 371 p^{56} T^{30} - 6 p^{61} T^{31} + p^{64} T^{32} \) | |
| 7 | \( 1 - 150 T + 11285 T^{2} - 105340 p T^{3} + 53104948 T^{4} - 3492716750 T^{5} + 210265419620 T^{6} - 13464566836550 T^{7} + 118440479336104 p T^{8} - 47039507839148380 T^{9} + 2720605987919130935 T^{10} - \)\(15\!\cdots\!50\)\( T^{11} + \)\(81\!\cdots\!71\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(16\!\cdots\!00\)\( p T^{15} + \)\(55\!\cdots\!80\)\( T^{16} - \)\(16\!\cdots\!00\)\( p^{5} T^{17} + \)\(22\!\cdots\!00\)\( p^{8} T^{18} - \)\(42\!\cdots\!00\)\( p^{12} T^{19} + \)\(81\!\cdots\!71\)\( p^{16} T^{20} - \)\(15\!\cdots\!50\)\( p^{20} T^{21} + 2720605987919130935 p^{24} T^{22} - 47039507839148380 p^{28} T^{23} + 118440479336104 p^{33} T^{24} - 13464566836550 p^{36} T^{25} + 210265419620 p^{40} T^{26} - 3492716750 p^{44} T^{27} + 53104948 p^{48} T^{28} - 105340 p^{53} T^{29} + 11285 p^{56} T^{30} - 150 p^{60} T^{31} + p^{64} T^{32} \) | |
| 13 | \( 1 + 510 T + 14709 p T^{2} + 62409260 T^{3} + 1371686832 p T^{4} + 4612357931190 T^{5} + 1115903207098688 T^{6} + 255861493025046550 T^{7} + 55129135753284721196 T^{8} + \)\(11\!\cdots\!80\)\( T^{9} + \)\(22\!\cdots\!31\)\( T^{10} + \)\(44\!\cdots\!10\)\( T^{11} + \)\(83\!\cdots\!19\)\( T^{12} + \)\(11\!\cdots\!60\)\( p T^{13} + \)\(27\!\cdots\!24\)\( T^{14} + \)\(47\!\cdots\!40\)\( T^{15} + \)\(80\!\cdots\!52\)\( T^{16} + \)\(47\!\cdots\!40\)\( p^{4} T^{17} + \)\(27\!\cdots\!24\)\( p^{8} T^{18} + \)\(11\!\cdots\!60\)\( p^{13} T^{19} + \)\(83\!\cdots\!19\)\( p^{16} T^{20} + \)\(44\!\cdots\!10\)\( p^{20} T^{21} + \)\(22\!\cdots\!31\)\( p^{24} T^{22} + \)\(11\!\cdots\!80\)\( p^{28} T^{23} + 55129135753284721196 p^{32} T^{24} + 255861493025046550 p^{36} T^{25} + 1115903207098688 p^{40} T^{26} + 4612357931190 p^{44} T^{27} + 1371686832 p^{49} T^{28} + 62409260 p^{52} T^{29} + 14709 p^{57} T^{30} + 510 p^{60} T^{31} + p^{64} T^{32} \) | |
| 17 | \( 1 - 1770 T + 1861377 T^{2} - 1465887360 T^{3} + 940402319088 T^{4} - 515277608889150 T^{5} + 246991922957083356 T^{6} - \)\(10\!\cdots\!30\)\( T^{7} + \)\(40\!\cdots\!32\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{9} + \)\(41\!\cdots\!95\)\( T^{10} - \)\(11\!\cdots\!50\)\( T^{11} + \)\(25\!\cdots\!47\)\( T^{12} - \)\(44\!\cdots\!60\)\( T^{13} + \)\(46\!\cdots\!72\)\( T^{14} + \)\(46\!\cdots\!80\)\( T^{15} - \)\(32\!\cdots\!60\)\( T^{16} + \)\(46\!\cdots\!80\)\( p^{4} T^{17} + \)\(46\!\cdots\!72\)\( p^{8} T^{18} - \)\(44\!\cdots\!60\)\( p^{12} T^{19} + \)\(25\!\cdots\!47\)\( p^{16} T^{20} - \)\(11\!\cdots\!50\)\( p^{20} T^{21} + \)\(41\!\cdots\!95\)\( p^{24} T^{22} - \)\(13\!\cdots\!40\)\( p^{28} T^{23} + \)\(40\!\cdots\!32\)\( p^{32} T^{24} - \)\(10\!\cdots\!30\)\( p^{36} T^{25} + 246991922957083356 p^{40} T^{26} - 515277608889150 p^{44} T^{27} + 940402319088 p^{48} T^{28} - 1465887360 p^{52} T^{29} + 1861377 p^{56} T^{30} - 1770 p^{60} T^{31} + p^{64} T^{32} \) | |
| 19 | \( 1 - 1020 T + 520302 T^{2} - 341536360 T^{3} + 226458128451 T^{4} - 108558942804660 T^{5} + 49666430293827668 T^{6} - 24904996025377771880 T^{7} + \)\(58\!\cdots\!69\)\( p T^{8} - \)\(45\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!56\)\( T^{10} - \)\(83\!\cdots\!20\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{13} + \)\(50\!\cdots\!54\)\( T^{14} - \)\(18\!\cdots\!00\)\( T^{15} + \)\(66\!\cdots\!12\)\( T^{16} - \)\(18\!\cdots\!00\)\( p^{4} T^{17} + \)\(50\!\cdots\!54\)\( p^{8} T^{18} - \)\(12\!\cdots\!20\)\( p^{12} T^{19} + \)\(32\!\cdots\!69\)\( p^{16} T^{20} - \)\(83\!\cdots\!20\)\( p^{20} T^{21} + \)\(19\!\cdots\!56\)\( p^{24} T^{22} - \)\(45\!\cdots\!00\)\( p^{28} T^{23} + \)\(58\!\cdots\!69\)\( p^{33} T^{24} - 24904996025377771880 p^{36} T^{25} + 49666430293827668 p^{40} T^{26} - 108558942804660 p^{44} T^{27} + 226458128451 p^{48} T^{28} - 341536360 p^{52} T^{29} + 520302 p^{56} T^{30} - 1020 p^{60} T^{31} + p^{64} T^{32} \) | |
| 23 | \( ( 1 + 1212 T + 1799148 T^{2} + 55776972 p T^{3} + 1162112305972 T^{4} + 637290145542300 T^{5} + 486872939194611412 T^{6} + \)\(23\!\cdots\!32\)\( T^{7} + \)\(15\!\cdots\!90\)\( T^{8} + \)\(23\!\cdots\!32\)\( p^{4} T^{9} + 486872939194611412 p^{8} T^{10} + 637290145542300 p^{12} T^{11} + 1162112305972 p^{16} T^{12} + 55776972 p^{21} T^{13} + 1799148 p^{24} T^{14} + 1212 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 29 | \( 1 - 4890 T + 13421633 T^{2} - 29067952860 T^{3} + 54052604988544 T^{4} - 88679737904682330 T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!30\)\( T^{7} + \)\(23\!\cdots\!88\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!35\)\( T^{10} - \)\(34\!\cdots\!70\)\( T^{11} + \)\(35\!\cdots\!31\)\( T^{12} - \)\(34\!\cdots\!60\)\( T^{13} + \)\(32\!\cdots\!32\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!88\)\( T^{16} - \)\(29\!\cdots\!00\)\( p^{4} T^{17} + \)\(32\!\cdots\!32\)\( p^{8} T^{18} - \)\(34\!\cdots\!60\)\( p^{12} T^{19} + \)\(35\!\cdots\!31\)\( p^{16} T^{20} - \)\(34\!\cdots\!70\)\( p^{20} T^{21} + \)\(31\!\cdots\!35\)\( p^{24} T^{22} - \)\(27\!\cdots\!00\)\( p^{28} T^{23} + \)\(23\!\cdots\!88\)\( p^{32} T^{24} - \)\(18\!\cdots\!30\)\( p^{36} T^{25} + \)\(13\!\cdots\!00\)\( p^{40} T^{26} - 88679737904682330 p^{44} T^{27} + 54052604988544 p^{48} T^{28} - 29067952860 p^{52} T^{29} + 13421633 p^{56} T^{30} - 4890 p^{60} T^{31} + p^{64} T^{32} \) | |
| 31 | \( 1 - 602 T - 3399183 T^{2} + 2476834452 T^{3} + 5090346936760 T^{4} - 5476528868607730 T^{5} - 3636052207408894608 T^{6} + \)\(84\!\cdots\!26\)\( T^{7} - \)\(11\!\cdots\!88\)\( T^{8} - \)\(96\!\cdots\!28\)\( T^{9} + \)\(63\!\cdots\!99\)\( T^{10} + \)\(87\!\cdots\!86\)\( T^{11} - \)\(92\!\cdots\!49\)\( T^{12} - \)\(58\!\cdots\!04\)\( T^{13} + \)\(95\!\cdots\!52\)\( T^{14} + \)\(19\!\cdots\!80\)\( T^{15} - \)\(88\!\cdots\!12\)\( T^{16} + \)\(19\!\cdots\!80\)\( p^{4} T^{17} + \)\(95\!\cdots\!52\)\( p^{8} T^{18} - \)\(58\!\cdots\!04\)\( p^{12} T^{19} - \)\(92\!\cdots\!49\)\( p^{16} T^{20} + \)\(87\!\cdots\!86\)\( p^{20} T^{21} + \)\(63\!\cdots\!99\)\( p^{24} T^{22} - \)\(96\!\cdots\!28\)\( p^{28} T^{23} - \)\(11\!\cdots\!88\)\( p^{32} T^{24} + \)\(84\!\cdots\!26\)\( p^{36} T^{25} - 3636052207408894608 p^{40} T^{26} - 5476528868607730 p^{44} T^{27} + 5090346936760 p^{48} T^{28} + 2476834452 p^{52} T^{29} - 3399183 p^{56} T^{30} - 602 p^{60} T^{31} + p^{64} T^{32} \) | |
| 37 | \( 1 + 4518 T + 5699745 T^{2} - 1900578312 T^{3} - 8549703770976 T^{4} - 9257557958642118 T^{5} - 19217512078487116564 T^{6} - \)\(19\!\cdots\!14\)\( T^{7} + \)\(19\!\cdots\!32\)\( T^{8} + \)\(46\!\cdots\!72\)\( T^{9} + \)\(49\!\cdots\!35\)\( T^{10} + \)\(27\!\cdots\!14\)\( p T^{11} + \)\(10\!\cdots\!07\)\( T^{12} - \)\(36\!\cdots\!44\)\( T^{13} - \)\(90\!\cdots\!36\)\( T^{14} - \)\(28\!\cdots\!20\)\( p T^{15} - \)\(22\!\cdots\!32\)\( T^{16} - \)\(28\!\cdots\!20\)\( p^{5} T^{17} - \)\(90\!\cdots\!36\)\( p^{8} T^{18} - \)\(36\!\cdots\!44\)\( p^{12} T^{19} + \)\(10\!\cdots\!07\)\( p^{16} T^{20} + \)\(27\!\cdots\!14\)\( p^{21} T^{21} + \)\(49\!\cdots\!35\)\( p^{24} T^{22} + \)\(46\!\cdots\!72\)\( p^{28} T^{23} + \)\(19\!\cdots\!32\)\( p^{32} T^{24} - \)\(19\!\cdots\!14\)\( p^{36} T^{25} - 19217512078487116564 p^{40} T^{26} - 9257557958642118 p^{44} T^{27} - 8549703770976 p^{48} T^{28} - 1900578312 p^{52} T^{29} + 5699745 p^{56} T^{30} + 4518 p^{60} T^{31} + p^{64} T^{32} \) | |
| 41 | \( 1 - 1290 T + 7180721 T^{2} - 24597131280 T^{3} + 33234776342592 T^{4} - 126750273427742670 T^{5} + \)\(22\!\cdots\!64\)\( T^{6} - \)\(28\!\cdots\!10\)\( T^{7} + \)\(90\!\cdots\!76\)\( T^{8} - \)\(79\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!95\)\( T^{10} - \)\(34\!\cdots\!10\)\( T^{11} + \)\(88\!\cdots\!23\)\( T^{12} - \)\(76\!\cdots\!00\)\( T^{13} + \)\(88\!\cdots\!60\)\( T^{14} - \)\(42\!\cdots\!20\)\( T^{15} + \)\(42\!\cdots\!72\)\( T^{16} - \)\(42\!\cdots\!20\)\( p^{4} T^{17} + \)\(88\!\cdots\!60\)\( p^{8} T^{18} - \)\(76\!\cdots\!00\)\( p^{12} T^{19} + \)\(88\!\cdots\!23\)\( p^{16} T^{20} - \)\(34\!\cdots\!10\)\( p^{20} T^{21} + \)\(14\!\cdots\!95\)\( p^{24} T^{22} - \)\(79\!\cdots\!20\)\( p^{28} T^{23} + \)\(90\!\cdots\!76\)\( p^{32} T^{24} - \)\(28\!\cdots\!10\)\( p^{36} T^{25} + \)\(22\!\cdots\!64\)\( p^{40} T^{26} - 126750273427742670 p^{44} T^{27} + 33234776342592 p^{48} T^{28} - 24597131280 p^{52} T^{29} + 7180721 p^{56} T^{30} - 1290 p^{60} T^{31} + p^{64} T^{32} \) | |
| 43 | \( 1 - 20987262 T^{2} + 259339094563399 T^{4} - \)\(23\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!83\)\( T^{8} - \)\(94\!\cdots\!80\)\( T^{10} + \)\(46\!\cdots\!01\)\( T^{12} - \)\(19\!\cdots\!18\)\( T^{14} + \)\(72\!\cdots\!68\)\( T^{16} - \)\(19\!\cdots\!18\)\( p^{8} T^{18} + \)\(46\!\cdots\!01\)\( p^{16} T^{20} - \)\(94\!\cdots\!80\)\( p^{24} T^{22} + \)\(16\!\cdots\!83\)\( p^{32} T^{24} - \)\(23\!\cdots\!20\)\( p^{40} T^{26} + 259339094563399 p^{48} T^{28} - 20987262 p^{56} T^{30} + p^{64} T^{32} \) | |
| 47 | \( 1 - 642 T - 14533935 T^{2} + 13327717908 T^{3} + 66878912304504 T^{4} - 253575586665950538 T^{5} + \)\(12\!\cdots\!16\)\( T^{6} + \)\(20\!\cdots\!86\)\( T^{7} - \)\(29\!\cdots\!88\)\( T^{8} - \)\(35\!\cdots\!08\)\( T^{9} + \)\(22\!\cdots\!35\)\( T^{10} - \)\(46\!\cdots\!02\)\( T^{11} - \)\(76\!\cdots\!13\)\( T^{12} + \)\(29\!\cdots\!96\)\( T^{13} - \)\(37\!\cdots\!16\)\( T^{14} - \)\(55\!\cdots\!00\)\( T^{15} + \)\(43\!\cdots\!08\)\( T^{16} - \)\(55\!\cdots\!00\)\( p^{4} T^{17} - \)\(37\!\cdots\!16\)\( p^{8} T^{18} + \)\(29\!\cdots\!96\)\( p^{12} T^{19} - \)\(76\!\cdots\!13\)\( p^{16} T^{20} - \)\(46\!\cdots\!02\)\( p^{20} T^{21} + \)\(22\!\cdots\!35\)\( p^{24} T^{22} - \)\(35\!\cdots\!08\)\( p^{28} T^{23} - \)\(29\!\cdots\!88\)\( p^{32} T^{24} + \)\(20\!\cdots\!86\)\( p^{36} T^{25} + \)\(12\!\cdots\!16\)\( p^{40} T^{26} - 253575586665950538 p^{44} T^{27} + 66878912304504 p^{48} T^{28} + 13327717908 p^{52} T^{29} - 14533935 p^{56} T^{30} - 642 p^{60} T^{31} + p^{64} T^{32} \) | |
| 53 | \( 1 - 2598 T - 23282099 T^{2} + 16085006940 T^{3} + 5471901661220 p T^{4} + 530348332632801306 T^{5} - \)\(85\!\cdots\!68\)\( T^{6} - \)\(10\!\cdots\!34\)\( T^{7} - \)\(31\!\cdots\!80\)\( T^{8} + \)\(56\!\cdots\!80\)\( T^{9} + \)\(46\!\cdots\!35\)\( T^{10} + \)\(55\!\cdots\!10\)\( T^{11} - \)\(19\!\cdots\!65\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} - \)\(37\!\cdots\!60\)\( p T^{14} + \)\(42\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!60\)\( T^{16} + \)\(42\!\cdots\!00\)\( p^{4} T^{17} - \)\(37\!\cdots\!60\)\( p^{9} T^{18} - \)\(10\!\cdots\!20\)\( p^{12} T^{19} - \)\(19\!\cdots\!65\)\( p^{16} T^{20} + \)\(55\!\cdots\!10\)\( p^{20} T^{21} + \)\(46\!\cdots\!35\)\( p^{24} T^{22} + \)\(56\!\cdots\!80\)\( p^{28} T^{23} - \)\(31\!\cdots\!80\)\( p^{32} T^{24} - \)\(10\!\cdots\!34\)\( p^{36} T^{25} - \)\(85\!\cdots\!68\)\( p^{40} T^{26} + 530348332632801306 p^{44} T^{27} + 5471901661220 p^{49} T^{28} + 16085006940 p^{52} T^{29} - 23282099 p^{56} T^{30} - 2598 p^{60} T^{31} + p^{64} T^{32} \) | |
| 59 | \( 1 - 6660 T + 22992662 T^{2} - 18655523280 T^{3} - 189806228621629 T^{4} + 494250155838373140 T^{5} + \)\(18\!\cdots\!28\)\( T^{6} - \)\(80\!\cdots\!60\)\( T^{7} - \)\(21\!\cdots\!89\)\( T^{8} + \)\(28\!\cdots\!20\)\( T^{9} - \)\(10\!\cdots\!64\)\( T^{10} + \)\(81\!\cdots\!00\)\( T^{11} + \)\(63\!\cdots\!49\)\( T^{12} - \)\(26\!\cdots\!20\)\( T^{13} + \)\(74\!\cdots\!34\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(63\!\cdots\!52\)\( T^{16} - \)\(19\!\cdots\!40\)\( p^{4} T^{17} + \)\(74\!\cdots\!34\)\( p^{8} T^{18} - \)\(26\!\cdots\!20\)\( p^{12} T^{19} + \)\(63\!\cdots\!49\)\( p^{16} T^{20} + \)\(81\!\cdots\!00\)\( p^{20} T^{21} - \)\(10\!\cdots\!64\)\( p^{24} T^{22} + \)\(28\!\cdots\!20\)\( p^{28} T^{23} - \)\(21\!\cdots\!89\)\( p^{32} T^{24} - \)\(80\!\cdots\!60\)\( p^{36} T^{25} + \)\(18\!\cdots\!28\)\( p^{40} T^{26} + 494250155838373140 p^{44} T^{27} - 189806228621629 p^{48} T^{28} - 18655523280 p^{52} T^{29} + 22992662 p^{56} T^{30} - 6660 p^{60} T^{31} + p^{64} T^{32} \) | |
| 61 | \( 1 + 27410 T + 400835965 T^{2} + 3993584493540 T^{3} + 29722963699190756 T^{4} + \)\(17\!\cdots\!10\)\( T^{5} + \)\(76\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!50\)\( T^{7} + \)\(31\!\cdots\!60\)\( T^{8} - \)\(20\!\cdots\!40\)\( T^{9} - \)\(19\!\cdots\!05\)\( T^{10} - \)\(94\!\cdots\!30\)\( T^{11} - \)\(29\!\cdots\!01\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!20\)\( T^{15} + \)\(91\!\cdots\!24\)\( T^{16} + \)\(20\!\cdots\!20\)\( p^{4} T^{17} + \)\(22\!\cdots\!00\)\( p^{8} T^{18} - \)\(38\!\cdots\!40\)\( p^{12} T^{19} - \)\(29\!\cdots\!01\)\( p^{16} T^{20} - \)\(94\!\cdots\!30\)\( p^{20} T^{21} - \)\(19\!\cdots\!05\)\( p^{24} T^{22} - \)\(20\!\cdots\!40\)\( p^{28} T^{23} + \)\(31\!\cdots\!60\)\( p^{32} T^{24} + \)\(23\!\cdots\!50\)\( p^{36} T^{25} + \)\(76\!\cdots\!00\)\( p^{40} T^{26} + \)\(17\!\cdots\!10\)\( p^{44} T^{27} + 29722963699190756 p^{48} T^{28} + 3993584493540 p^{52} T^{29} + 400835965 p^{56} T^{30} + 27410 p^{60} T^{31} + p^{64} T^{32} \) | |
| 67 | \( ( 1 - 10762 T + 159918251 T^{2} - 1314523445328 T^{3} + 11214987232726071 T^{4} - 72053371844041564716 T^{5} + \)\(45\!\cdots\!49\)\( T^{6} - \)\(23\!\cdots\!74\)\( T^{7} + \)\(11\!\cdots\!72\)\( T^{8} - \)\(23\!\cdots\!74\)\( p^{4} T^{9} + \)\(45\!\cdots\!49\)\( p^{8} T^{10} - 72053371844041564716 p^{12} T^{11} + 11214987232726071 p^{16} T^{12} - 1314523445328 p^{20} T^{13} + 159918251 p^{24} T^{14} - 10762 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 71 | \( 1 + 5562 T - 19466523 T^{2} - 418912559172 T^{3} - 1153762191456540 T^{4} - 2108818435709926950 T^{5} + \)\(29\!\cdots\!92\)\( T^{6} + \)\(31\!\cdots\!14\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} - \)\(21\!\cdots\!12\)\( T^{9} - \)\(33\!\cdots\!61\)\( T^{10} - \)\(21\!\cdots\!46\)\( T^{11} - \)\(21\!\cdots\!49\)\( T^{12} - \)\(17\!\cdots\!76\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} + \)\(13\!\cdots\!60\)\( T^{15} + \)\(26\!\cdots\!68\)\( T^{16} + \)\(13\!\cdots\!60\)\( p^{4} T^{17} + \)\(19\!\cdots\!12\)\( p^{8} T^{18} - \)\(17\!\cdots\!76\)\( p^{12} T^{19} - \)\(21\!\cdots\!49\)\( p^{16} T^{20} - \)\(21\!\cdots\!46\)\( p^{20} T^{21} - \)\(33\!\cdots\!61\)\( p^{24} T^{22} - \)\(21\!\cdots\!12\)\( p^{28} T^{23} + \)\(21\!\cdots\!12\)\( p^{32} T^{24} + \)\(31\!\cdots\!14\)\( p^{36} T^{25} + \)\(29\!\cdots\!92\)\( p^{40} T^{26} - 2108818435709926950 p^{44} T^{27} - 1153762191456540 p^{48} T^{28} - 418912559172 p^{52} T^{29} - 19466523 p^{56} T^{30} + 5562 p^{60} T^{31} + p^{64} T^{32} \) | |
| 73 | \( 1 + 7790 T + 144123241 T^{2} + 948598487200 T^{3} + 8890043553940480 T^{4} + 48987457404564962410 T^{5} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{7} + \)\(61\!\cdots\!20\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!23\)\( T^{10} + \)\(51\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!43\)\( T^{12} + \)\(93\!\cdots\!00\)\( T^{13} + \)\(99\!\cdots\!00\)\( T^{14} + \)\(38\!\cdots\!20\)\( T^{15} + \)\(42\!\cdots\!60\)\( T^{16} + \)\(38\!\cdots\!20\)\( p^{4} T^{17} + \)\(99\!\cdots\!00\)\( p^{8} T^{18} + \)\(93\!\cdots\!00\)\( p^{12} T^{19} + \)\(15\!\cdots\!43\)\( p^{16} T^{20} + \)\(51\!\cdots\!70\)\( p^{20} T^{21} + \)\(82\!\cdots\!23\)\( p^{24} T^{22} + \)\(33\!\cdots\!00\)\( p^{28} T^{23} + \)\(61\!\cdots\!20\)\( p^{32} T^{24} + \)\(15\!\cdots\!10\)\( p^{36} T^{25} + \)\(30\!\cdots\!80\)\( p^{40} T^{26} + 48987457404564962410 p^{44} T^{27} + 8890043553940480 p^{48} T^{28} + 948598487200 p^{52} T^{29} + 144123241 p^{56} T^{30} + 7790 p^{60} T^{31} + p^{64} T^{32} \) | |
| 79 | \( 1 + 2770 T + 166665437 T^{2} + 526138782040 T^{3} + 12136795940958028 T^{4} + 43837435641583269910 T^{5} + \)\(42\!\cdots\!16\)\( T^{6} + \)\(16\!\cdots\!90\)\( T^{7} + \)\(76\!\cdots\!32\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} - \)\(46\!\cdots\!25\)\( T^{10} - \)\(13\!\cdots\!50\)\( T^{11} - \)\(27\!\cdots\!93\)\( T^{12} + \)\(44\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!92\)\( T^{14} + \)\(76\!\cdots\!60\)\( T^{15} + \)\(93\!\cdots\!00\)\( T^{16} + \)\(76\!\cdots\!60\)\( p^{4} T^{17} + \)\(15\!\cdots\!92\)\( p^{8} T^{18} + \)\(44\!\cdots\!00\)\( p^{12} T^{19} - \)\(27\!\cdots\!93\)\( p^{16} T^{20} - \)\(13\!\cdots\!50\)\( p^{20} T^{21} - \)\(46\!\cdots\!25\)\( p^{24} T^{22} + \)\(17\!\cdots\!00\)\( p^{28} T^{23} + \)\(76\!\cdots\!32\)\( p^{32} T^{24} + \)\(16\!\cdots\!90\)\( p^{36} T^{25} + \)\(42\!\cdots\!16\)\( p^{40} T^{26} + 43837435641583269910 p^{44} T^{27} + 12136795940958028 p^{48} T^{28} + 526138782040 p^{52} T^{29} + 166665437 p^{56} T^{30} + 2770 p^{60} T^{31} + p^{64} T^{32} \) | |
| 83 | \( 1 + 36900 T + 763618438 T^{2} + 11718168958800 T^{3} + 150953322079676179 T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!23\)\( T^{8} + \)\(15\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(85\!\cdots\!81\)\( T^{12} + \)\(66\!\cdots\!80\)\( T^{13} + \)\(49\!\cdots\!82\)\( T^{14} + \)\(35\!\cdots\!20\)\( T^{15} + \)\(24\!\cdots\!88\)\( T^{16} + \)\(35\!\cdots\!20\)\( p^{4} T^{17} + \)\(49\!\cdots\!82\)\( p^{8} T^{18} + \)\(66\!\cdots\!80\)\( p^{12} T^{19} + \)\(85\!\cdots\!81\)\( p^{16} T^{20} + \)\(10\!\cdots\!40\)\( p^{20} T^{21} + \)\(13\!\cdots\!80\)\( p^{24} T^{22} + \)\(15\!\cdots\!40\)\( p^{28} T^{23} + \)\(17\!\cdots\!23\)\( p^{32} T^{24} + \)\(18\!\cdots\!40\)\( p^{36} T^{25} + \)\(18\!\cdots\!20\)\( p^{40} T^{26} + \)\(17\!\cdots\!00\)\( p^{44} T^{27} + 150953322079676179 p^{48} T^{28} + 11718168958800 p^{52} T^{29} + 763618438 p^{56} T^{30} + 36900 p^{60} T^{31} + p^{64} T^{32} \) | |
| 89 | \( ( 1 - 23298 T + 382639319 T^{2} - 4171807048380 T^{3} + 42200999392839103 T^{4} - \)\(36\!\cdots\!60\)\( T^{5} + \)\(29\!\cdots\!21\)\( T^{6} - \)\(20\!\cdots\!22\)\( T^{7} + \)\(15\!\cdots\!48\)\( T^{8} - \)\(20\!\cdots\!22\)\( p^{4} T^{9} + \)\(29\!\cdots\!21\)\( p^{8} T^{10} - \)\(36\!\cdots\!60\)\( p^{12} T^{11} + 42200999392839103 p^{16} T^{12} - 4171807048380 p^{20} T^{13} + 382639319 p^{24} T^{14} - 23298 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 97 | \( 1 + 3732 T - 207823798 T^{2} - 2405333600772 T^{3} + 24453132413373135 T^{4} + \)\(45\!\cdots\!00\)\( T^{5} - \)\(14\!\cdots\!88\)\( T^{6} - \)\(65\!\cdots\!36\)\( T^{7} - \)\(30\!\cdots\!13\)\( T^{8} + \)\(73\!\cdots\!88\)\( T^{9} + \)\(32\!\cdots\!64\)\( T^{10} - \)\(68\!\cdots\!56\)\( T^{11} - \)\(56\!\cdots\!99\)\( T^{12} + \)\(42\!\cdots\!24\)\( T^{13} + \)\(69\!\cdots\!62\)\( T^{14} - \)\(13\!\cdots\!40\)\( T^{15} - \)\(66\!\cdots\!52\)\( T^{16} - \)\(13\!\cdots\!40\)\( p^{4} T^{17} + \)\(69\!\cdots\!62\)\( p^{8} T^{18} + \)\(42\!\cdots\!24\)\( p^{12} T^{19} - \)\(56\!\cdots\!99\)\( p^{16} T^{20} - \)\(68\!\cdots\!56\)\( p^{20} T^{21} + \)\(32\!\cdots\!64\)\( p^{24} T^{22} + \)\(73\!\cdots\!88\)\( p^{28} T^{23} - \)\(30\!\cdots\!13\)\( p^{32} T^{24} - \)\(65\!\cdots\!36\)\( p^{36} T^{25} - \)\(14\!\cdots\!88\)\( p^{40} T^{26} + \)\(45\!\cdots\!00\)\( p^{44} T^{27} + 24453132413373135 p^{48} T^{28} - 2405333600772 p^{52} T^{29} - 207823798 p^{56} T^{30} + 3732 p^{60} T^{31} + p^{64} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.21301964670799711798814230634, −5.18620625857112749101609176906, −5.10576407421415761283526258435, −5.03088502405639105164031106135, −4.85877282448112001423328539298, −4.81067997770020638707388062760, −4.52607543903145141929038022420, −4.33449392873367146407264188241, −4.10083018289100502167020880873, −4.03434484492506444843614661199, −4.01349774258012008038434699937, −3.90902538062603142194163316340, −3.30570143357465197730561399986, −2.95344681081298942812920248758, −2.93675260341134483727610053522, −2.83075108727346017896975635341, −2.81377302853264878004155944041, −2.77256955430456118478933495881, −1.99479565140039701177443395087, −1.83144922761744357141495034490, −1.60500214129071352138658893838, −1.35965912786835398253891122184, −1.30367413813979631455360695801, −1.04828963252608992778989220748, −0.76020164479704840167870810603, 0.76020164479704840167870810603, 1.04828963252608992778989220748, 1.30367413813979631455360695801, 1.35965912786835398253891122184, 1.60500214129071352138658893838, 1.83144922761744357141495034490, 1.99479565140039701177443395087, 2.77256955430456118478933495881, 2.81377302853264878004155944041, 2.83075108727346017896975635341, 2.93675260341134483727610053522, 2.95344681081298942812920248758, 3.30570143357465197730561399986, 3.90902538062603142194163316340, 4.01349774258012008038434699937, 4.03434484492506444843614661199, 4.10083018289100502167020880873, 4.33449392873367146407264188241, 4.52607543903145141929038022420, 4.81067997770020638707388062760, 4.85877282448112001423328539298, 5.03088502405639105164031106135, 5.10576407421415761283526258435, 5.18620625857112749101609176906, 5.21301964670799711798814230634
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.