Dirichlet series
| L(s) = 1 | + 4·5-s + 9·9-s + 5·11-s + 5·17-s − 15·19-s − 12·23-s + 17·25-s + 20·29-s − 3·31-s + 2·37-s − 4·41-s + 22·43-s + 36·45-s − 15·47-s − 28·49-s + 25·53-s + 20·55-s − 8·59-s − 46·61-s + 45·67-s − 15·71-s + 34·73-s + 45·81-s − 12·83-s + 20·85-s − 60·95-s − 40·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 3·9-s + 1.50·11-s + 1.21·17-s − 3.44·19-s − 2.50·23-s + 17/5·25-s + 3.71·29-s − 0.538·31-s + 0.328·37-s − 0.624·41-s + 3.35·43-s + 5.36·45-s − 2.18·47-s − 4·49-s + 3.43·53-s + 2.69·55-s − 1.04·59-s − 5.88·61-s + 5.49·67-s − 1.78·71-s + 3.97·73-s + 5·81-s − 1.31·83-s + 2.16·85-s − 6.15·95-s − 4.06·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 41^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 41^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.21288\times 10^{11}\) |
| Root analytic conductor: | \(2.28870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 41^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(66.20863577\) |
| \(L(\frac12)\) | \(\approx\) | \(66.20863577\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + 4 T - 8 T^{2} - 39 T^{3} + 781 T^{4} - 1780 T^{5} - 9077 T^{6} + 235635 T^{7} + 4795542 T^{8} + 235635 p T^{9} - 9077 p^{2} T^{10} - 1780 p^{3} T^{11} + 781 p^{4} T^{12} - 39 p^{5} T^{13} - 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 - p^{2} T^{2} + 4 p^{2} T^{4} - 5 p^{3} T^{6} + 653 T^{8} - 305 p^{2} T^{10} + 352 p^{3} T^{12} - 3311 p^{2} T^{14} + 89788 T^{16} - 3311 p^{4} T^{18} + 352 p^{7} T^{20} - 305 p^{8} T^{22} + 653 p^{8} T^{24} - 5 p^{13} T^{26} + 4 p^{14} T^{28} - p^{16} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 - 4 T - T^{2} + 41 T^{3} - 69 T^{4} - 37 T^{5} + 96 T^{6} - 317 T^{7} + 3602 T^{8} - 6078 T^{9} - 17079 T^{10} + 49592 T^{11} + 19471 T^{12} - 38671 T^{13} - 342361 T^{14} - 274826 T^{15} + 3399336 T^{16} - 274826 p T^{17} - 342361 p^{2} T^{18} - 38671 p^{3} T^{19} + 19471 p^{4} T^{20} + 49592 p^{5} T^{21} - 17079 p^{6} T^{22} - 6078 p^{7} T^{23} + 3602 p^{8} T^{24} - 317 p^{9} T^{25} + 96 p^{10} T^{26} - 37 p^{11} T^{27} - 69 p^{12} T^{28} + 41 p^{13} T^{29} - p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 4 p T^{2} - 20 T^{3} + 8 p^{2} T^{4} - 80 p T^{5} + 4512 T^{6} - 7740 T^{7} + 47317 T^{8} - 83495 T^{9} + 418400 T^{10} - 832150 T^{11} + 3293403 T^{12} - 7196995 T^{13} + 3589773 p T^{14} - 53516060 T^{15} + 182846400 T^{16} - 53516060 p T^{17} + 3589773 p^{3} T^{18} - 7196995 p^{3} T^{19} + 3293403 p^{4} T^{20} - 832150 p^{5} T^{21} + 418400 p^{6} T^{22} - 83495 p^{7} T^{23} + 47317 p^{8} T^{24} - 7740 p^{9} T^{25} + 4512 p^{10} T^{26} - 80 p^{12} T^{27} + 8 p^{14} T^{28} - 20 p^{13} T^{29} + 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 5 T + 63 T^{2} - 420 T^{3} + 2568 T^{4} - 16205 T^{5} + 82849 T^{6} - 427280 T^{7} + 2047567 T^{8} - 9076645 T^{9} + 39590230 T^{10} - 14706905 p T^{11} + 634554422 T^{12} - 219695845 p T^{13} + 8751803303 T^{14} - 30653196935 T^{15} + 104022058530 T^{16} - 30653196935 p T^{17} + 8751803303 p^{2} T^{18} - 219695845 p^{4} T^{19} + 634554422 p^{4} T^{20} - 14706905 p^{6} T^{21} + 39590230 p^{6} T^{22} - 9076645 p^{7} T^{23} + 2047567 p^{8} T^{24} - 427280 p^{9} T^{25} + 82849 p^{10} T^{26} - 16205 p^{11} T^{27} + 2568 p^{12} T^{28} - 420 p^{13} T^{29} + 63 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 27 T^{2} - 150 T^{3} + 542 T^{4} - 4050 T^{5} + 20713 T^{6} - 82305 T^{7} + 440287 T^{8} - 1975395 T^{9} + 7781680 T^{10} - 2701100 p T^{11} + 11147036 p T^{12} - 531445435 T^{13} + 2179765819 T^{14} - 8325043825 T^{15} + 27709757390 T^{16} - 8325043825 p T^{17} + 2179765819 p^{2} T^{18} - 531445435 p^{3} T^{19} + 11147036 p^{5} T^{20} - 2701100 p^{6} T^{21} + 7781680 p^{6} T^{22} - 1975395 p^{7} T^{23} + 440287 p^{8} T^{24} - 82305 p^{9} T^{25} + 20713 p^{10} T^{26} - 4050 p^{11} T^{27} + 542 p^{12} T^{28} - 150 p^{13} T^{29} + 27 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 5 T + 63 T^{2} - 25 p T^{3} + 2568 T^{4} - 16395 T^{5} + 78638 T^{6} - 439920 T^{7} + 2015796 T^{8} - 9574165 T^{9} + 46403810 T^{10} - 202652735 T^{11} + 971098702 T^{12} - 4240819090 T^{13} + 18882484980 T^{14} - 82602641445 T^{15} + 334049922892 T^{16} - 82602641445 p T^{17} + 18882484980 p^{2} T^{18} - 4240819090 p^{3} T^{19} + 971098702 p^{4} T^{20} - 202652735 p^{5} T^{21} + 46403810 p^{6} T^{22} - 9574165 p^{7} T^{23} + 2015796 p^{8} T^{24} - 439920 p^{9} T^{25} + 78638 p^{10} T^{26} - 16395 p^{11} T^{27} + 2568 p^{12} T^{28} - 25 p^{14} T^{29} + 63 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 15 T + 161 T^{2} + 1525 T^{3} + 597 p T^{4} + 75230 T^{5} + 443166 T^{6} + 2254670 T^{7} + 544273 p T^{8} + 39337075 T^{9} + 112992035 T^{10} + 121397655 T^{11} - 1613776893 T^{12} - 16094021990 T^{13} - 105965524802 T^{14} - 571838637400 T^{15} - 2626157417260 T^{16} - 571838637400 p T^{17} - 105965524802 p^{2} T^{18} - 16094021990 p^{3} T^{19} - 1613776893 p^{4} T^{20} + 121397655 p^{5} T^{21} + 112992035 p^{6} T^{22} + 39337075 p^{7} T^{23} + 544273 p^{9} T^{24} + 2254670 p^{9} T^{25} + 443166 p^{10} T^{26} + 75230 p^{11} T^{27} + 597 p^{13} T^{28} + 1525 p^{13} T^{29} + 161 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 12 T + 19 T^{2} - 157 T^{3} + 373 T^{4} + 373 T^{5} - 55496 T^{6} - 151147 T^{7} + 538774 T^{8} - 101222 T^{9} - 13084983 T^{10} + 28574544 T^{11} + 649978935 T^{12} + 1623339109 T^{13} - 493963175 p T^{14} - 27845846104 T^{15} + 180405296180 T^{16} - 27845846104 p T^{17} - 493963175 p^{3} T^{18} + 1623339109 p^{3} T^{19} + 649978935 p^{4} T^{20} + 28574544 p^{5} T^{21} - 13084983 p^{6} T^{22} - 101222 p^{7} T^{23} + 538774 p^{8} T^{24} - 151147 p^{9} T^{25} - 55496 p^{10} T^{26} + 373 p^{11} T^{27} + 373 p^{12} T^{28} - 157 p^{13} T^{29} + 19 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 20 T + 262 T^{2} - 2865 T^{3} + 26553 T^{4} - 218810 T^{5} + 1634626 T^{6} - 11342245 T^{7} + 74202867 T^{8} - 467497335 T^{9} + 2884176435 T^{10} - 17480048660 T^{11} + 105365484402 T^{12} - 627344579160 T^{13} + 3654811098552 T^{14} - 20729749116985 T^{15} + 113615735370380 T^{16} - 20729749116985 p T^{17} + 3654811098552 p^{2} T^{18} - 627344579160 p^{3} T^{19} + 105365484402 p^{4} T^{20} - 17480048660 p^{5} T^{21} + 2884176435 p^{6} T^{22} - 467497335 p^{7} T^{23} + 74202867 p^{8} T^{24} - 11342245 p^{9} T^{25} + 1634626 p^{10} T^{26} - 218810 p^{11} T^{27} + 26553 p^{12} T^{28} - 2865 p^{13} T^{29} + 262 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 3 T - 58 T^{2} - 298 T^{3} + 2045 T^{4} + 9320 T^{5} - 84713 T^{6} - 195904 T^{7} + 4105617 T^{8} + 3872962 T^{9} - 208627856 T^{10} - 219047229 T^{11} + 7596781121 T^{12} + 10652941876 T^{13} - 215328917398 T^{14} - 143234049680 T^{15} + 6349226698338 T^{16} - 143234049680 p T^{17} - 215328917398 p^{2} T^{18} + 10652941876 p^{3} T^{19} + 7596781121 p^{4} T^{20} - 219047229 p^{5} T^{21} - 208627856 p^{6} T^{22} + 3872962 p^{7} T^{23} + 4105617 p^{8} T^{24} - 195904 p^{9} T^{25} - 84713 p^{10} T^{26} + 9320 p^{11} T^{27} + 2045 p^{12} T^{28} - 298 p^{13} T^{29} - 58 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 2 T - 2 p T^{2} + 678 T^{3} - 780 T^{4} - 29004 T^{5} + 378159 T^{6} - 1092324 T^{7} - 11455407 T^{8} + 127580274 T^{9} - 441840318 T^{10} - 1264690140 T^{11} + 30961324455 T^{12} - 4881647226 p T^{13} + 138027863589 T^{14} + 5063066119648 T^{15} - 41354097034082 T^{16} + 5063066119648 p T^{17} + 138027863589 p^{2} T^{18} - 4881647226 p^{4} T^{19} + 30961324455 p^{4} T^{20} - 1264690140 p^{5} T^{21} - 441840318 p^{6} T^{22} + 127580274 p^{7} T^{23} - 11455407 p^{8} T^{24} - 1092324 p^{9} T^{25} + 378159 p^{10} T^{26} - 29004 p^{11} T^{27} - 780 p^{12} T^{28} + 678 p^{13} T^{29} - 2 p^{15} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 22 T + 79 T^{2} + 1977 T^{3} - 21187 T^{4} - 36273 T^{5} + 1842424 T^{6} - 7386453 T^{7} - 81497976 T^{8} + 823659492 T^{9} + 946182427 T^{10} - 48714667904 T^{11} + 150258950255 T^{12} + 1852630882821 T^{13} - 14387507087295 T^{14} - 31395815128266 T^{15} + 753378404697000 T^{16} - 31395815128266 p T^{17} - 14387507087295 p^{2} T^{18} + 1852630882821 p^{3} T^{19} + 150258950255 p^{4} T^{20} - 48714667904 p^{5} T^{21} + 946182427 p^{6} T^{22} + 823659492 p^{7} T^{23} - 81497976 p^{8} T^{24} - 7386453 p^{9} T^{25} + 1842424 p^{10} T^{26} - 36273 p^{11} T^{27} - 21187 p^{12} T^{28} + 1977 p^{13} T^{29} + 79 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 15 T + 319 T^{2} + 2235 T^{3} + 21716 T^{4} - 65835 T^{5} - 1013420 T^{6} - 23744670 T^{7} - 101995022 T^{8} - 14738925 p T^{9} + 5996295850 T^{10} + 52425012705 T^{11} + 657222543244 T^{12} + 1547403488760 T^{13} + 1776164108904 T^{14} - 186789793556565 T^{15} - 1153542114136992 T^{16} - 186789793556565 p T^{17} + 1776164108904 p^{2} T^{18} + 1547403488760 p^{3} T^{19} + 657222543244 p^{4} T^{20} + 52425012705 p^{5} T^{21} + 5996295850 p^{6} T^{22} - 14738925 p^{8} T^{23} - 101995022 p^{8} T^{24} - 23744670 p^{9} T^{25} - 1013420 p^{10} T^{26} - 65835 p^{11} T^{27} + 21716 p^{12} T^{28} + 2235 p^{13} T^{29} + 319 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 25 T + 413 T^{2} - 4205 T^{3} + 32887 T^{4} - 166490 T^{5} + 817862 T^{6} - 4406590 T^{7} + 65399227 T^{8} - 632468245 T^{9} + 5998127695 T^{10} - 35889696115 T^{11} + 268218227003 T^{12} - 1658499250930 T^{13} + 16763296531886 T^{14} - 122805038494960 T^{15} + 1088882590000420 T^{16} - 122805038494960 p T^{17} + 16763296531886 p^{2} T^{18} - 1658499250930 p^{3} T^{19} + 268218227003 p^{4} T^{20} - 35889696115 p^{5} T^{21} + 5998127695 p^{6} T^{22} - 632468245 p^{7} T^{23} + 65399227 p^{8} T^{24} - 4406590 p^{9} T^{25} + 817862 p^{10} T^{26} - 166490 p^{11} T^{27} + 32887 p^{12} T^{28} - 4205 p^{13} T^{29} + 413 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 8 T + 15 T^{2} + 857 T^{3} + 6687 T^{4} + 41163 T^{5} + 671648 T^{6} + 2868903 T^{7} + 8726994 T^{8} + 173530180 T^{9} + 350103221 T^{10} - 1362124186 T^{11} - 23336798021 T^{12} - 942522742505 T^{13} - 6933616153029 T^{14} - 46988570559288 T^{15} - 517997837841536 T^{16} - 46988570559288 p T^{17} - 6933616153029 p^{2} T^{18} - 942522742505 p^{3} T^{19} - 23336798021 p^{4} T^{20} - 1362124186 p^{5} T^{21} + 350103221 p^{6} T^{22} + 173530180 p^{7} T^{23} + 8726994 p^{8} T^{24} + 2868903 p^{9} T^{25} + 671648 p^{10} T^{26} + 41163 p^{11} T^{27} + 6687 p^{12} T^{28} + 857 p^{13} T^{29} + 15 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 46 T + 1087 T^{2} + 17449 T^{3} + 208527 T^{4} + 1850343 T^{5} + 10881866 T^{6} + 17261585 T^{7} - 460530370 T^{8} - 6071458282 T^{9} - 36969498781 T^{10} - 32534160406 T^{11} + 1562486034213 T^{12} + 13365571910049 T^{13} + 222707996563 T^{14} - 1080062594520004 T^{15} - 12289078565267476 T^{16} - 1080062594520004 p T^{17} + 222707996563 p^{2} T^{18} + 13365571910049 p^{3} T^{19} + 1562486034213 p^{4} T^{20} - 32534160406 p^{5} T^{21} - 36969498781 p^{6} T^{22} - 6071458282 p^{7} T^{23} - 460530370 p^{8} T^{24} + 17261585 p^{9} T^{25} + 10881866 p^{10} T^{26} + 1850343 p^{11} T^{27} + 208527 p^{12} T^{28} + 17449 p^{13} T^{29} + 1087 p^{14} T^{30} + 46 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 45 T + 971 T^{2} - 15715 T^{3} + 242357 T^{4} - 3491360 T^{5} + 44996702 T^{6} - 544413470 T^{7} + 6325993107 T^{8} - 69410222595 T^{9} + 725476220135 T^{10} - 7307681959045 T^{11} + 70480990733133 T^{12} - 653690636722980 T^{13} + 5867787032194624 T^{14} - 50626271793942110 T^{15} + 420725817995597460 T^{16} - 50626271793942110 p T^{17} + 5867787032194624 p^{2} T^{18} - 653690636722980 p^{3} T^{19} + 70480990733133 p^{4} T^{20} - 7307681959045 p^{5} T^{21} + 725476220135 p^{6} T^{22} - 69410222595 p^{7} T^{23} + 6325993107 p^{8} T^{24} - 544413470 p^{9} T^{25} + 44996702 p^{10} T^{26} - 3491360 p^{11} T^{27} + 242357 p^{12} T^{28} - 15715 p^{13} T^{29} + 971 p^{14} T^{30} - 45 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 15 T + 333 T^{2} + 2115 T^{3} + 23468 T^{4} - 110895 T^{5} - 564886 T^{6} - 27387070 T^{7} - 18538168 T^{8} - 1092956195 T^{9} + 13634357530 T^{10} + 12169056285 T^{11} + 1382715900342 T^{12} - 2624673612340 T^{13} + 23918358394978 T^{14} - 1010473424039345 T^{15} - 3122417250610940 T^{16} - 1010473424039345 p T^{17} + 23918358394978 p^{2} T^{18} - 2624673612340 p^{3} T^{19} + 1382715900342 p^{4} T^{20} + 12169056285 p^{5} T^{21} + 13634357530 p^{6} T^{22} - 1092956195 p^{7} T^{23} - 18538168 p^{8} T^{24} - 27387070 p^{9} T^{25} - 564886 p^{10} T^{26} - 110895 p^{11} T^{27} + 23468 p^{12} T^{28} + 2115 p^{13} T^{29} + 333 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 17 T + 485 T^{2} - 5868 T^{3} + 100534 T^{4} - 986446 T^{5} + 12875724 T^{6} - 106236101 T^{7} + 1127476896 T^{8} - 106236101 p T^{9} + 12875724 p^{2} T^{10} - 986446 p^{3} T^{11} + 100534 p^{4} T^{12} - 5868 p^{5} T^{13} + 485 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 737 T^{2} + 270304 T^{4} - 65719015 T^{6} + 11895746573 T^{8} - 1706613469745 T^{10} + 201629278078916 T^{12} - 20096005908122503 T^{14} + 1712905283658948508 T^{16} - 20096005908122503 p^{2} T^{18} + 201629278078916 p^{4} T^{20} - 1706613469745 p^{6} T^{22} + 11895746573 p^{8} T^{24} - 65719015 p^{10} T^{26} + 270304 p^{12} T^{28} - 737 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 6 T + 471 T^{2} + 3054 T^{3} + 107213 T^{4} + 687396 T^{5} + 15437769 T^{6} + 89913372 T^{7} + 1529216388 T^{8} + 89913372 p T^{9} + 15437769 p^{2} T^{10} + 687396 p^{3} T^{11} + 107213 p^{4} T^{12} + 3054 p^{5} T^{13} + 471 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 225 T^{2} + 980 T^{3} + 20690 T^{4} + 220500 T^{5} + 1287265 T^{6} + 31278295 T^{7} + 74234149 T^{8} + 3462955545 T^{9} + 13523503080 T^{10} + 281191192280 T^{11} + 2332233990710 T^{12} + 19198263573455 T^{13} + 290452934135005 T^{14} + 987525869517405 T^{15} + 29870868329825326 T^{16} + 987525869517405 p T^{17} + 290452934135005 p^{2} T^{18} + 19198263573455 p^{3} T^{19} + 2332233990710 p^{4} T^{20} + 281191192280 p^{5} T^{21} + 13523503080 p^{6} T^{22} + 3462955545 p^{7} T^{23} + 74234149 p^{8} T^{24} + 31278295 p^{9} T^{25} + 1287265 p^{10} T^{26} + 220500 p^{11} T^{27} + 20690 p^{12} T^{28} + 980 p^{13} T^{29} + 225 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 40 T + 1103 T^{2} + 23400 T^{3} + 417832 T^{4} + 6531120 T^{5} + 91158672 T^{6} + 1150615280 T^{7} + 13185683112 T^{8} + 138351028200 T^{9} + 1326877723405 T^{10} + 120031839960 p T^{11} + 92843992326243 T^{12} + 669334303943600 T^{13} + 4412489859579696 T^{14} + 28605503736845280 T^{15} + 230828447985637600 T^{16} + 28605503736845280 p T^{17} + 4412489859579696 p^{2} T^{18} + 669334303943600 p^{3} T^{19} + 92843992326243 p^{4} T^{20} + 120031839960 p^{6} T^{21} + 1326877723405 p^{6} T^{22} + 138351028200 p^{7} T^{23} + 13185683112 p^{8} T^{24} + 1150615280 p^{9} T^{25} + 91158672 p^{10} T^{26} + 6531120 p^{11} T^{27} + 417832 p^{12} T^{28} + 23400 p^{13} T^{29} + 1103 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.79436575755939392329937708692, −2.68127194965846445686144167151, −2.62748326666741076866664685025, −2.51698066636569463029012155937, −2.42562906130422530115751635700, −2.40782156333169309789098321814, −2.29337930165138745339380847505, −2.15398428600813468677865147034, −2.01296035478910673984570122919, −2.01281809596048268807933882692, −1.91958005329854514756719524196, −1.87324222857515587447801759084, −1.81532673458062338321889640824, −1.59921314000114323075122951611, −1.54737163440000949194291951313, −1.47603708989790013488506654884, −1.45940036371139770322055619364, −1.24469776008030190553320084670, −1.22316430787157760853922300022, −1.00950116348536694618911565334, −0.967197781801331603562718837141, −0.64602406153790044597415792572, −0.54547251641647547617993036445, −0.46230083932868000141159933358, −0.43823437015706999107409834704, 0.43823437015706999107409834704, 0.46230083932868000141159933358, 0.54547251641647547617993036445, 0.64602406153790044597415792572, 0.967197781801331603562718837141, 1.00950116348536694618911565334, 1.22316430787157760853922300022, 1.24469776008030190553320084670, 1.45940036371139770322055619364, 1.47603708989790013488506654884, 1.54737163440000949194291951313, 1.59921314000114323075122951611, 1.81532673458062338321889640824, 1.87324222857515587447801759084, 1.91958005329854514756719524196, 2.01281809596048268807933882692, 2.01296035478910673984570122919, 2.15398428600813468677865147034, 2.29337930165138745339380847505, 2.40782156333169309789098321814, 2.42562906130422530115751635700, 2.51698066636569463029012155937, 2.62748326666741076866664685025, 2.68127194965846445686144167151, 2.79436575755939392329937708692
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.