Dirichlet series
| L(s) = 1 | − 4·2-s + 48·3-s − 20·4-s − 192·6-s + 90·8-s + 1.22e3·9-s − 176·11-s − 960·12-s − 104·13-s + 135·16-s − 220·17-s − 4.89e3·18-s − 152·19-s + 704·22-s − 180·23-s + 4.32e3·24-s − 858·25-s + 416·26-s + 2.20e4·27-s − 604·29-s − 380·31-s − 708·32-s − 8.44e3·33-s + 880·34-s − 2.44e4·36-s + 148·37-s + 608·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 9.23·3-s − 5/2·4-s − 13.0·6-s + 3.97·8-s + 45.3·9-s − 4.82·11-s − 23.0·12-s − 2.21·13-s + 2.10·16-s − 3.13·17-s − 64.1·18-s − 1.83·19-s + 6.82·22-s − 1.63·23-s + 36.7·24-s − 6.86·25-s + 3.13·26-s + 157.·27-s − 3.86·29-s − 2.20·31-s − 3.91·32-s − 44.5·33-s + 4.43·34-s − 113.·36-s + 0.657·37-s + 2.59·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 7^{32} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.71213\times 10^{31}\) |
| Root analytic conductor: | \(9.76760\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 3^{16} \cdot 7^{32} \cdot 11^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{16} \) |
| 7 | \( 1 \) | |
| 11 | \( ( 1 + p T )^{16} \) | |
| good | 2 | \( 1 + p^{2} T + 9 p^{2} T^{2} + 67 p T^{3} + 761 T^{4} + 663 p^{2} T^{5} + 11997 T^{6} + 19835 p T^{7} + 77355 p T^{8} + 122661 p^{2} T^{9} + 1744271 T^{10} + 331199 p^{4} T^{11} + 1097357 p^{4} T^{12} + 1606281 p^{5} T^{13} + 10043213 p^{4} T^{14} + 883093 p^{9} T^{15} + 5259293 p^{8} T^{16} + 883093 p^{12} T^{17} + 10043213 p^{10} T^{18} + 1606281 p^{14} T^{19} + 1097357 p^{16} T^{20} + 331199 p^{19} T^{21} + 1744271 p^{18} T^{22} + 122661 p^{23} T^{23} + 77355 p^{25} T^{24} + 19835 p^{28} T^{25} + 11997 p^{30} T^{26} + 663 p^{35} T^{27} + 761 p^{36} T^{28} + 67 p^{40} T^{29} + 9 p^{44} T^{30} + p^{47} T^{31} + p^{48} T^{32} \) |
| 5 | \( 1 + 858 T^{2} + 508 T^{3} + 354306 T^{4} + 374816 T^{5} + 94222194 T^{6} + 156113344 T^{7} + 3641534437 p T^{8} + 50875143788 T^{9} + 2760285036784 T^{10} + 543041489924 p^{2} T^{11} + 353640459581174 T^{12} + 2868479067027584 T^{13} + 41936343455918148 T^{14} + 93647546801597332 p T^{15} + 5071505396739663388 T^{16} + 93647546801597332 p^{4} T^{17} + 41936343455918148 p^{6} T^{18} + 2868479067027584 p^{9} T^{19} + 353640459581174 p^{12} T^{20} + 543041489924 p^{17} T^{21} + 2760285036784 p^{18} T^{22} + 50875143788 p^{21} T^{23} + 3641534437 p^{25} T^{24} + 156113344 p^{27} T^{25} + 94222194 p^{30} T^{26} + 374816 p^{33} T^{27} + 354306 p^{36} T^{28} + 508 p^{39} T^{29} + 858 p^{42} T^{30} + p^{48} T^{32} \) | |
| 13 | \( 1 + 8 p T + 20164 T^{2} + 1608816 T^{3} + 185020014 T^{4} + 12318317624 T^{5} + 1090529611696 T^{6} + 63862530337656 T^{7} + 4815368614616577 T^{8} + 256511830052541032 T^{9} + 17289123464530634280 T^{10} + \)\(85\!\cdots\!64\)\( T^{11} + \)\(52\!\cdots\!70\)\( T^{12} + \)\(18\!\cdots\!64\)\( p T^{13} + \)\(13\!\cdots\!92\)\( T^{14} + \)\(60\!\cdots\!40\)\( T^{15} + \)\(32\!\cdots\!00\)\( T^{16} + \)\(60\!\cdots\!40\)\( p^{3} T^{17} + \)\(13\!\cdots\!92\)\( p^{6} T^{18} + \)\(18\!\cdots\!64\)\( p^{10} T^{19} + \)\(52\!\cdots\!70\)\( p^{12} T^{20} + \)\(85\!\cdots\!64\)\( p^{15} T^{21} + 17289123464530634280 p^{18} T^{22} + 256511830052541032 p^{21} T^{23} + 4815368614616577 p^{24} T^{24} + 63862530337656 p^{27} T^{25} + 1090529611696 p^{30} T^{26} + 12318317624 p^{33} T^{27} + 185020014 p^{36} T^{28} + 1608816 p^{39} T^{29} + 20164 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32} \) | |
| 17 | \( 1 + 220 T + 62686 T^{2} + 9991148 T^{3} + 1698630592 T^{4} + 216421003540 T^{5} + 27889215137490 T^{6} + 3005128944778212 T^{7} + 321999941246354540 T^{8} + 30500790174878449068 T^{9} + \)\(28\!\cdots\!58\)\( T^{10} + \)\(24\!\cdots\!16\)\( T^{11} + \)\(20\!\cdots\!12\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!94\)\( T^{14} + \)\(54\!\cdots\!52\)\( p T^{15} + \)\(39\!\cdots\!70\)\( p T^{16} + \)\(54\!\cdots\!52\)\( p^{4} T^{17} + \)\(12\!\cdots\!94\)\( p^{6} T^{18} + \)\(16\!\cdots\!60\)\( p^{9} T^{19} + \)\(20\!\cdots\!12\)\( p^{12} T^{20} + \)\(24\!\cdots\!16\)\( p^{15} T^{21} + \)\(28\!\cdots\!58\)\( p^{18} T^{22} + 30500790174878449068 p^{21} T^{23} + 321999941246354540 p^{24} T^{24} + 3005128944778212 p^{27} T^{25} + 27889215137490 p^{30} T^{26} + 216421003540 p^{33} T^{27} + 1698630592 p^{36} T^{28} + 9991148 p^{39} T^{29} + 62686 p^{42} T^{30} + 220 p^{45} T^{31} + p^{48} T^{32} \) | |
| 19 | \( 1 + 8 p T + 52610 T^{2} + 297860 p T^{3} + 1215708974 T^{4} + 100939472224 T^{5} + 17822615327262 T^{6} + 1170958981753192 T^{7} + 193423342840421097 T^{8} + 9862931601846933492 T^{9} + \)\(16\!\cdots\!76\)\( T^{10} + \)\(63\!\cdots\!84\)\( T^{11} + \)\(12\!\cdots\!58\)\( T^{12} + \)\(33\!\cdots\!20\)\( T^{13} + \)\(89\!\cdots\!04\)\( T^{14} + \)\(16\!\cdots\!12\)\( T^{15} + \)\(61\!\cdots\!92\)\( T^{16} + \)\(16\!\cdots\!12\)\( p^{3} T^{17} + \)\(89\!\cdots\!04\)\( p^{6} T^{18} + \)\(33\!\cdots\!20\)\( p^{9} T^{19} + \)\(12\!\cdots\!58\)\( p^{12} T^{20} + \)\(63\!\cdots\!84\)\( p^{15} T^{21} + \)\(16\!\cdots\!76\)\( p^{18} T^{22} + 9862931601846933492 p^{21} T^{23} + 193423342840421097 p^{24} T^{24} + 1170958981753192 p^{27} T^{25} + 17822615327262 p^{30} T^{26} + 100939472224 p^{33} T^{27} + 1215708974 p^{36} T^{28} + 297860 p^{40} T^{29} + 52610 p^{42} T^{30} + 8 p^{46} T^{31} + p^{48} T^{32} \) | |
| 23 | \( 1 + 180 T + 84320 T^{2} + 10494100 T^{3} + 2862760004 T^{4} + 11172527164 p T^{5} + 56791492579472 T^{6} + 4048145397719044 T^{7} + 904513920275301444 T^{8} + 66300323956050152436 T^{9} + \)\(14\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(23\!\cdots\!08\)\( T^{12} + \)\(17\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!88\)\( T^{14} + \)\(20\!\cdots\!64\)\( T^{15} + \)\(36\!\cdots\!70\)\( T^{16} + \)\(20\!\cdots\!64\)\( p^{3} T^{17} + \)\(30\!\cdots\!88\)\( p^{6} T^{18} + \)\(17\!\cdots\!68\)\( p^{9} T^{19} + \)\(23\!\cdots\!08\)\( p^{12} T^{20} + \)\(11\!\cdots\!60\)\( p^{15} T^{21} + \)\(14\!\cdots\!92\)\( p^{18} T^{22} + 66300323956050152436 p^{21} T^{23} + 904513920275301444 p^{24} T^{24} + 4048145397719044 p^{27} T^{25} + 56791492579472 p^{30} T^{26} + 11172527164 p^{34} T^{27} + 2862760004 p^{36} T^{28} + 10494100 p^{39} T^{29} + 84320 p^{42} T^{30} + 180 p^{45} T^{31} + p^{48} T^{32} \) | |
| 29 | \( 1 + 604 T + 360024 T^{2} + 142843872 T^{3} + 53189735950 T^{4} + 16288126944204 T^{5} + 4694702294215672 T^{6} + 1193285511081551916 T^{7} + \)\(28\!\cdots\!17\)\( T^{8} + \)\(63\!\cdots\!88\)\( T^{9} + \)\(13\!\cdots\!12\)\( T^{10} + \)\(26\!\cdots\!56\)\( T^{11} + \)\(49\!\cdots\!18\)\( T^{12} + \)\(88\!\cdots\!84\)\( T^{13} + \)\(15\!\cdots\!32\)\( T^{14} + \)\(24\!\cdots\!68\)\( T^{15} + \)\(40\!\cdots\!44\)\( T^{16} + \)\(24\!\cdots\!68\)\( p^{3} T^{17} + \)\(15\!\cdots\!32\)\( p^{6} T^{18} + \)\(88\!\cdots\!84\)\( p^{9} T^{19} + \)\(49\!\cdots\!18\)\( p^{12} T^{20} + \)\(26\!\cdots\!56\)\( p^{15} T^{21} + \)\(13\!\cdots\!12\)\( p^{18} T^{22} + \)\(63\!\cdots\!88\)\( p^{21} T^{23} + \)\(28\!\cdots\!17\)\( p^{24} T^{24} + 1193285511081551916 p^{27} T^{25} + 4694702294215672 p^{30} T^{26} + 16288126944204 p^{33} T^{27} + 53189735950 p^{36} T^{28} + 142843872 p^{39} T^{29} + 360024 p^{42} T^{30} + 604 p^{45} T^{31} + p^{48} T^{32} \) | |
| 31 | \( 1 + 380 T + 271138 T^{2} + 84289740 T^{3} + 35587622964 T^{4} + 9317506110124 T^{5} + 2981555481221590 T^{6} + 672211192221141276 T^{7} + \)\(17\!\cdots\!96\)\( T^{8} + \)\(35\!\cdots\!12\)\( T^{9} + \)\(81\!\cdots\!70\)\( T^{10} + \)\(14\!\cdots\!88\)\( T^{11} + \)\(30\!\cdots\!04\)\( T^{12} + \)\(48\!\cdots\!64\)\( T^{13} + \)\(98\!\cdots\!10\)\( T^{14} + \)\(15\!\cdots\!96\)\( T^{15} + \)\(29\!\cdots\!78\)\( T^{16} + \)\(15\!\cdots\!96\)\( p^{3} T^{17} + \)\(98\!\cdots\!10\)\( p^{6} T^{18} + \)\(48\!\cdots\!64\)\( p^{9} T^{19} + \)\(30\!\cdots\!04\)\( p^{12} T^{20} + \)\(14\!\cdots\!88\)\( p^{15} T^{21} + \)\(81\!\cdots\!70\)\( p^{18} T^{22} + \)\(35\!\cdots\!12\)\( p^{21} T^{23} + \)\(17\!\cdots\!96\)\( p^{24} T^{24} + 672211192221141276 p^{27} T^{25} + 2981555481221590 p^{30} T^{26} + 9317506110124 p^{33} T^{27} + 35587622964 p^{36} T^{28} + 84289740 p^{39} T^{29} + 271138 p^{42} T^{30} + 380 p^{45} T^{31} + p^{48} T^{32} \) | |
| 37 | \( 1 - 4 p T + 393496 T^{2} - 73492252 T^{3} + 80823509014 T^{4} - 17372290877452 T^{5} + 11438293063834460 T^{6} - 2674565435595493572 T^{7} + \)\(12\!\cdots\!33\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{9} + \)\(10\!\cdots\!16\)\( T^{10} - \)\(27\!\cdots\!28\)\( T^{11} + \)\(80\!\cdots\!14\)\( T^{12} - \)\(19\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!48\)\( T^{14} - \)\(11\!\cdots\!88\)\( T^{15} + \)\(27\!\cdots\!36\)\( T^{16} - \)\(11\!\cdots\!88\)\( p^{3} T^{17} + \)\(50\!\cdots\!48\)\( p^{6} T^{18} - \)\(19\!\cdots\!40\)\( p^{9} T^{19} + \)\(80\!\cdots\!14\)\( p^{12} T^{20} - \)\(27\!\cdots\!28\)\( p^{15} T^{21} + \)\(10\!\cdots\!16\)\( p^{18} T^{22} - \)\(30\!\cdots\!92\)\( p^{21} T^{23} + \)\(12\!\cdots\!33\)\( p^{24} T^{24} - 2674565435595493572 p^{27} T^{25} + 11438293063834460 p^{30} T^{26} - 17372290877452 p^{33} T^{27} + 80823509014 p^{36} T^{28} - 73492252 p^{39} T^{29} + 393496 p^{42} T^{30} - 4 p^{46} T^{31} + p^{48} T^{32} \) | |
| 41 | \( 1 + 60 T + 764634 T^{2} + 57429452 T^{3} + 290989198348 T^{4} + 24444526283748 T^{5} + 72768497388466270 T^{6} + 6433370889780806068 T^{7} + \)\(13\!\cdots\!16\)\( T^{8} + \)\(11\!\cdots\!32\)\( T^{9} + \)\(18\!\cdots\!38\)\( T^{10} + \)\(16\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!60\)\( T^{12} + \)\(17\!\cdots\!56\)\( T^{13} + \)\(19\!\cdots\!62\)\( T^{14} + \)\(15\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!06\)\( T^{16} + \)\(15\!\cdots\!24\)\( p^{3} T^{17} + \)\(19\!\cdots\!62\)\( p^{6} T^{18} + \)\(17\!\cdots\!56\)\( p^{9} T^{19} + \)\(21\!\cdots\!60\)\( p^{12} T^{20} + \)\(16\!\cdots\!40\)\( p^{15} T^{21} + \)\(18\!\cdots\!38\)\( p^{18} T^{22} + \)\(11\!\cdots\!32\)\( p^{21} T^{23} + \)\(13\!\cdots\!16\)\( p^{24} T^{24} + 6433370889780806068 p^{27} T^{25} + 72768497388466270 p^{30} T^{26} + 24444526283748 p^{33} T^{27} + 290989198348 p^{36} T^{28} + 57429452 p^{39} T^{29} + 764634 p^{42} T^{30} + 60 p^{45} T^{31} + p^{48} T^{32} \) | |
| 43 | \( 1 - 252 T + 703624 T^{2} - 203054604 T^{3} + 248992647172 T^{4} - 76614651093708 T^{5} + 58908851011328040 T^{6} - 426324017975976404 p T^{7} + \)\(10\!\cdots\!72\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{9} + \)\(14\!\cdots\!32\)\( T^{10} - \)\(42\!\cdots\!52\)\( T^{11} + \)\(16\!\cdots\!80\)\( T^{12} - \)\(46\!\cdots\!36\)\( T^{13} + \)\(16\!\cdots\!40\)\( T^{14} - \)\(43\!\cdots\!12\)\( T^{15} + \)\(14\!\cdots\!98\)\( T^{16} - \)\(43\!\cdots\!12\)\( p^{3} T^{17} + \)\(16\!\cdots\!40\)\( p^{6} T^{18} - \)\(46\!\cdots\!36\)\( p^{9} T^{19} + \)\(16\!\cdots\!80\)\( p^{12} T^{20} - \)\(42\!\cdots\!52\)\( p^{15} T^{21} + \)\(14\!\cdots\!32\)\( p^{18} T^{22} - \)\(31\!\cdots\!88\)\( p^{21} T^{23} + \)\(10\!\cdots\!72\)\( p^{24} T^{24} - 426324017975976404 p^{28} T^{25} + 58908851011328040 p^{30} T^{26} - 76614651093708 p^{33} T^{27} + 248992647172 p^{36} T^{28} - 203054604 p^{39} T^{29} + 703624 p^{42} T^{30} - 252 p^{45} T^{31} + p^{48} T^{32} \) | |
| 47 | \( 1 + 1468 T + 1914870 T^{2} + 1642171092 T^{3} + 1299850600894 T^{4} + 832080174583100 T^{5} + 505710717798305182 T^{6} + \)\(26\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!13\)\( T^{8} + \)\(62\!\cdots\!60\)\( T^{9} + \)\(28\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!08\)\( T^{11} + \)\(46\!\cdots\!94\)\( T^{12} + \)\(17\!\cdots\!24\)\( T^{13} + \)\(62\!\cdots\!64\)\( T^{14} + \)\(21\!\cdots\!68\)\( T^{15} + \)\(70\!\cdots\!80\)\( T^{16} + \)\(21\!\cdots\!68\)\( p^{3} T^{17} + \)\(62\!\cdots\!64\)\( p^{6} T^{18} + \)\(17\!\cdots\!24\)\( p^{9} T^{19} + \)\(46\!\cdots\!94\)\( p^{12} T^{20} + \)\(11\!\cdots\!08\)\( p^{15} T^{21} + \)\(28\!\cdots\!40\)\( p^{18} T^{22} + \)\(62\!\cdots\!60\)\( p^{21} T^{23} + \)\(13\!\cdots\!13\)\( p^{24} T^{24} + \)\(26\!\cdots\!80\)\( p^{27} T^{25} + 505710717798305182 p^{30} T^{26} + 832080174583100 p^{33} T^{27} + 1299850600894 p^{36} T^{28} + 1642171092 p^{39} T^{29} + 1914870 p^{42} T^{30} + 1468 p^{45} T^{31} + p^{48} T^{32} \) | |
| 53 | \( 1 + 1456 T + 1679084 T^{2} + 1220008864 T^{3} + 14914705516 p T^{4} + 400007151737120 T^{5} + 207668104886404068 T^{6} + 96199875028054690640 T^{7} + \)\(49\!\cdots\!48\)\( T^{8} + \)\(21\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!56\)\( T^{10} + \)\(77\!\cdots\!60\)\( p T^{11} + \)\(17\!\cdots\!84\)\( T^{12} + \)\(69\!\cdots\!08\)\( T^{13} + \)\(56\!\cdots\!96\)\( p T^{14} + \)\(11\!\cdots\!12\)\( T^{15} + \)\(47\!\cdots\!38\)\( T^{16} + \)\(11\!\cdots\!12\)\( p^{3} T^{17} + \)\(56\!\cdots\!96\)\( p^{7} T^{18} + \)\(69\!\cdots\!08\)\( p^{9} T^{19} + \)\(17\!\cdots\!84\)\( p^{12} T^{20} + \)\(77\!\cdots\!60\)\( p^{16} T^{21} + \)\(10\!\cdots\!56\)\( p^{18} T^{22} + \)\(21\!\cdots\!52\)\( p^{21} T^{23} + \)\(49\!\cdots\!48\)\( p^{24} T^{24} + 96199875028054690640 p^{27} T^{25} + 207668104886404068 p^{30} T^{26} + 400007151737120 p^{33} T^{27} + 14914705516 p^{37} T^{28} + 1220008864 p^{39} T^{29} + 1679084 p^{42} T^{30} + 1456 p^{45} T^{31} + p^{48} T^{32} \) | |
| 59 | \( 1 + 1312 T + 1721688 T^{2} + 1551367400 T^{3} + 1295524255862 T^{4} + 938568010503848 T^{5} + 624515591319946400 T^{6} + \)\(38\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!53\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{9} + \)\(67\!\cdots\!20\)\( T^{10} + \)\(34\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!42\)\( T^{12} + \)\(88\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!60\)\( T^{15} + \)\(92\!\cdots\!80\)\( T^{16} + \)\(20\!\cdots\!60\)\( p^{3} T^{17} + \)\(42\!\cdots\!96\)\( p^{6} T^{18} + \)\(88\!\cdots\!28\)\( p^{9} T^{19} + \)\(17\!\cdots\!42\)\( p^{12} T^{20} + \)\(34\!\cdots\!20\)\( p^{15} T^{21} + \)\(67\!\cdots\!20\)\( p^{18} T^{22} + \)\(12\!\cdots\!12\)\( p^{21} T^{23} + \)\(22\!\cdots\!53\)\( p^{24} T^{24} + \)\(38\!\cdots\!60\)\( p^{27} T^{25} + 624515591319946400 p^{30} T^{26} + 938568010503848 p^{33} T^{27} + 1295524255862 p^{36} T^{28} + 1551367400 p^{39} T^{29} + 1721688 p^{42} T^{30} + 1312 p^{45} T^{31} + p^{48} T^{32} \) | |
| 61 | \( 1 + 2880 T + 4951064 T^{2} + 6179519168 T^{3} + 6185805793600 T^{4} + 5150467598399552 T^{5} + 3661345536223667016 T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!96\)\( T^{8} + \)\(52\!\cdots\!16\)\( T^{9} + \)\(18\!\cdots\!24\)\( T^{10} + \)\(45\!\cdots\!92\)\( T^{11} - \)\(66\!\cdots\!76\)\( T^{12} - \)\(95\!\cdots\!96\)\( T^{13} - \)\(75\!\cdots\!00\)\( T^{14} - \)\(43\!\cdots\!24\)\( T^{15} - \)\(22\!\cdots\!50\)\( T^{16} - \)\(43\!\cdots\!24\)\( p^{3} T^{17} - \)\(75\!\cdots\!00\)\( p^{6} T^{18} - \)\(95\!\cdots\!96\)\( p^{9} T^{19} - \)\(66\!\cdots\!76\)\( p^{12} T^{20} + \)\(45\!\cdots\!92\)\( p^{15} T^{21} + \)\(18\!\cdots\!24\)\( p^{18} T^{22} + \)\(52\!\cdots\!16\)\( p^{21} T^{23} + \)\(11\!\cdots\!96\)\( p^{24} T^{24} + \)\(22\!\cdots\!80\)\( p^{27} T^{25} + 3661345536223667016 p^{30} T^{26} + 5150467598399552 p^{33} T^{27} + 6185805793600 p^{36} T^{28} + 6179519168 p^{39} T^{29} + 4951064 p^{42} T^{30} + 2880 p^{45} T^{31} + p^{48} T^{32} \) | |
| 67 | \( 1 - 1220 T + 3169876 T^{2} - 3074096704 T^{3} + 4603456562494 T^{4} - 3764296673741828 T^{5} + 4219820102613756144 T^{6} - \)\(30\!\cdots\!36\)\( T^{7} + \)\(28\!\cdots\!01\)\( T^{8} - \)\(17\!\cdots\!72\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} - \)\(85\!\cdots\!04\)\( T^{11} + \)\(62\!\cdots\!82\)\( T^{12} - \)\(33\!\cdots\!84\)\( T^{13} + \)\(22\!\cdots\!56\)\( T^{14} - \)\(11\!\cdots\!24\)\( T^{15} + \)\(73\!\cdots\!44\)\( T^{16} - \)\(11\!\cdots\!24\)\( p^{3} T^{17} + \)\(22\!\cdots\!56\)\( p^{6} T^{18} - \)\(33\!\cdots\!84\)\( p^{9} T^{19} + \)\(62\!\cdots\!82\)\( p^{12} T^{20} - \)\(85\!\cdots\!04\)\( p^{15} T^{21} + \)\(14\!\cdots\!56\)\( p^{18} T^{22} - \)\(17\!\cdots\!72\)\( p^{21} T^{23} + \)\(28\!\cdots\!01\)\( p^{24} T^{24} - \)\(30\!\cdots\!36\)\( p^{27} T^{25} + 4219820102613756144 p^{30} T^{26} - 3764296673741828 p^{33} T^{27} + 4603456562494 p^{36} T^{28} - 3074096704 p^{39} T^{29} + 3169876 p^{42} T^{30} - 1220 p^{45} T^{31} + p^{48} T^{32} \) | |
| 71 | \( 1 + 2040 T + 4455488 T^{2} + 5620319112 T^{3} + 7277807393936 T^{4} + 6784275171138056 T^{5} + 6627521736470954016 T^{6} + \)\(49\!\cdots\!72\)\( T^{7} + \)\(41\!\cdots\!76\)\( T^{8} + \)\(26\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!84\)\( T^{11} + \)\(85\!\cdots\!48\)\( T^{12} + \)\(47\!\cdots\!92\)\( T^{13} + \)\(33\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!88\)\( T^{15} + \)\(12\!\cdots\!82\)\( T^{16} + \)\(17\!\cdots\!88\)\( p^{3} T^{17} + \)\(33\!\cdots\!96\)\( p^{6} T^{18} + \)\(47\!\cdots\!92\)\( p^{9} T^{19} + \)\(85\!\cdots\!48\)\( p^{12} T^{20} + \)\(11\!\cdots\!84\)\( p^{15} T^{21} + \)\(20\!\cdots\!68\)\( p^{18} T^{22} + \)\(26\!\cdots\!40\)\( p^{21} T^{23} + \)\(41\!\cdots\!76\)\( p^{24} T^{24} + \)\(49\!\cdots\!72\)\( p^{27} T^{25} + 6627521736470954016 p^{30} T^{26} + 6784275171138056 p^{33} T^{27} + 7277807393936 p^{36} T^{28} + 5620319112 p^{39} T^{29} + 4455488 p^{42} T^{30} + 2040 p^{45} T^{31} + p^{48} T^{32} \) | |
| 73 | \( 1 + 1628 T + 3195454 T^{2} + 3360488612 T^{3} + 4427024398930 T^{4} + 3878453778556828 T^{5} + 4224915737670222978 T^{6} + \)\(32\!\cdots\!44\)\( T^{7} + \)\(31\!\cdots\!01\)\( T^{8} + \)\(21\!\cdots\!64\)\( T^{9} + \)\(19\!\cdots\!76\)\( T^{10} + \)\(12\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!06\)\( T^{12} + \)\(61\!\cdots\!20\)\( T^{13} + \)\(47\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!08\)\( T^{15} + \)\(19\!\cdots\!44\)\( T^{16} + \)\(27\!\cdots\!08\)\( p^{3} T^{17} + \)\(47\!\cdots\!00\)\( p^{6} T^{18} + \)\(61\!\cdots\!20\)\( p^{9} T^{19} + \)\(10\!\cdots\!06\)\( p^{12} T^{20} + \)\(12\!\cdots\!48\)\( p^{15} T^{21} + \)\(19\!\cdots\!76\)\( p^{18} T^{22} + \)\(21\!\cdots\!64\)\( p^{21} T^{23} + \)\(31\!\cdots\!01\)\( p^{24} T^{24} + \)\(32\!\cdots\!44\)\( p^{27} T^{25} + 4224915737670222978 p^{30} T^{26} + 3878453778556828 p^{33} T^{27} + 4427024398930 p^{36} T^{28} + 3360488612 p^{39} T^{29} + 3195454 p^{42} T^{30} + 1628 p^{45} T^{31} + p^{48} T^{32} \) | |
| 79 | \( 1 + 416 T + 4601512 T^{2} + 2672083328 T^{3} + 10730093788232 T^{4} + 7470784069576512 T^{5} + 17056068187914917816 T^{6} + \)\(12\!\cdots\!76\)\( T^{7} + \)\(20\!\cdots\!80\)\( T^{8} + \)\(15\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!72\)\( T^{11} + \)\(15\!\cdots\!52\)\( T^{12} + \)\(10\!\cdots\!08\)\( T^{13} + \)\(10\!\cdots\!60\)\( T^{14} + \)\(63\!\cdots\!52\)\( T^{15} + \)\(55\!\cdots\!74\)\( T^{16} + \)\(63\!\cdots\!52\)\( p^{3} T^{17} + \)\(10\!\cdots\!60\)\( p^{6} T^{18} + \)\(10\!\cdots\!08\)\( p^{9} T^{19} + \)\(15\!\cdots\!52\)\( p^{12} T^{20} + \)\(14\!\cdots\!72\)\( p^{15} T^{21} + \)\(19\!\cdots\!16\)\( p^{18} T^{22} + \)\(15\!\cdots\!00\)\( p^{21} T^{23} + \)\(20\!\cdots\!80\)\( p^{24} T^{24} + \)\(12\!\cdots\!76\)\( p^{27} T^{25} + 17056068187914917816 p^{30} T^{26} + 7470784069576512 p^{33} T^{27} + 10730093788232 p^{36} T^{28} + 2672083328 p^{39} T^{29} + 4601512 p^{42} T^{30} + 416 p^{45} T^{31} + p^{48} T^{32} \) | |
| 83 | \( 1 + 3724 T + 10980954 T^{2} + 22243437596 T^{3} + 38404946640996 T^{4} + 53905197727917132 T^{5} + 66760024267616536302 T^{6} + \)\(71\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!40\)\( T^{8} + \)\(59\!\cdots\!04\)\( T^{9} + \)\(49\!\cdots\!70\)\( T^{10} + \)\(39\!\cdots\!28\)\( T^{11} + \)\(33\!\cdots\!00\)\( T^{12} + \)\(28\!\cdots\!12\)\( T^{13} + \)\(25\!\cdots\!74\)\( T^{14} + \)\(20\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!34\)\( T^{16} + \)\(20\!\cdots\!08\)\( p^{3} T^{17} + \)\(25\!\cdots\!74\)\( p^{6} T^{18} + \)\(28\!\cdots\!12\)\( p^{9} T^{19} + \)\(33\!\cdots\!00\)\( p^{12} T^{20} + \)\(39\!\cdots\!28\)\( p^{15} T^{21} + \)\(49\!\cdots\!70\)\( p^{18} T^{22} + \)\(59\!\cdots\!04\)\( p^{21} T^{23} + \)\(68\!\cdots\!40\)\( p^{24} T^{24} + \)\(71\!\cdots\!00\)\( p^{27} T^{25} + 66760024267616536302 p^{30} T^{26} + 53905197727917132 p^{33} T^{27} + 38404946640996 p^{36} T^{28} + 22243437596 p^{39} T^{29} + 10980954 p^{42} T^{30} + 3724 p^{45} T^{31} + p^{48} T^{32} \) | |
| 89 | \( 1 + 752 T + 73808 p T^{2} + 4408052176 T^{3} + 21318743091808 T^{4} + 12951025490619376 T^{5} + 45910381161538720496 T^{6} + \)\(25\!\cdots\!92\)\( T^{7} + \)\(74\!\cdots\!28\)\( T^{8} + \)\(38\!\cdots\!00\)\( T^{9} + \)\(95\!\cdots\!92\)\( T^{10} + \)\(46\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!60\)\( T^{12} + \)\(45\!\cdots\!04\)\( T^{13} + \)\(91\!\cdots\!36\)\( T^{14} + \)\(38\!\cdots\!40\)\( T^{15} + \)\(69\!\cdots\!38\)\( T^{16} + \)\(38\!\cdots\!40\)\( p^{3} T^{17} + \)\(91\!\cdots\!36\)\( p^{6} T^{18} + \)\(45\!\cdots\!04\)\( p^{9} T^{19} + \)\(10\!\cdots\!60\)\( p^{12} T^{20} + \)\(46\!\cdots\!48\)\( p^{15} T^{21} + \)\(95\!\cdots\!92\)\( p^{18} T^{22} + \)\(38\!\cdots\!00\)\( p^{21} T^{23} + \)\(74\!\cdots\!28\)\( p^{24} T^{24} + \)\(25\!\cdots\!92\)\( p^{27} T^{25} + 45910381161538720496 p^{30} T^{26} + 12951025490619376 p^{33} T^{27} + 21318743091808 p^{36} T^{28} + 4408052176 p^{39} T^{29} + 73808 p^{43} T^{30} + 752 p^{45} T^{31} + p^{48} T^{32} \) | |
| 97 | \( 1 + 1088 T + 3243796 T^{2} + 1931890048 T^{3} + 5415242615572 T^{4} + 2437970407787520 T^{5} + 8449944515556523260 T^{6} + \)\(33\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!64\)\( T^{8} + \)\(38\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} + \)\(40\!\cdots\!24\)\( T^{11} + \)\(15\!\cdots\!28\)\( T^{12} + \)\(40\!\cdots\!72\)\( T^{13} + \)\(16\!\cdots\!64\)\( T^{14} + \)\(38\!\cdots\!48\)\( T^{15} + \)\(15\!\cdots\!58\)\( T^{16} + \)\(38\!\cdots\!48\)\( p^{3} T^{17} + \)\(16\!\cdots\!64\)\( p^{6} T^{18} + \)\(40\!\cdots\!72\)\( p^{9} T^{19} + \)\(15\!\cdots\!28\)\( p^{12} T^{20} + \)\(40\!\cdots\!24\)\( p^{15} T^{21} + \)\(14\!\cdots\!48\)\( p^{18} T^{22} + \)\(38\!\cdots\!84\)\( p^{21} T^{23} + \)\(11\!\cdots\!64\)\( p^{24} T^{24} + \)\(33\!\cdots\!16\)\( p^{27} T^{25} + 8449944515556523260 p^{30} T^{26} + 2437970407787520 p^{33} T^{27} + 5415242615572 p^{36} T^{28} + 1931890048 p^{39} T^{29} + 3243796 p^{42} T^{30} + 1088 p^{45} T^{31} + p^{48} T^{32} \) | |
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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
−2.42304849285641513514818016302, −2.36472050741883705781292792626, −2.33512230506469217832493818583, −2.31737002505866217139330607921, −2.23768410938006552614946621067, −2.22882784808710970182081812103, −2.21812889766920245605260256247, −2.18632794480250948746434363449, −2.08578209500129275481962360134, −2.05554342746935608488702754912, −2.03210885974618049944283195420, −1.82674360579125943933669988774, −1.69223288140630265812525974921, −1.60801097823514929117880016144, −1.52755503227984962321556201650, −1.51881909215584452987354972102, −1.43401647643373906479189877676, −1.35517546366644694633545609953, −1.32691143235336719384527446645, −1.30185839663816882557466990766, −1.22193362571606263270272842193, −1.13788861865375632642342947746, −1.08397859958114096782435561083, −1.05222186326633301814709508398, −0.978893324969226094585264949181, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.978893324969226094585264949181, 1.05222186326633301814709508398, 1.08397859958114096782435561083, 1.13788861865375632642342947746, 1.22193362571606263270272842193, 1.30185839663816882557466990766, 1.32691143235336719384527446645, 1.35517546366644694633545609953, 1.43401647643373906479189877676, 1.51881909215584452987354972102, 1.52755503227984962321556201650, 1.60801097823514929117880016144, 1.69223288140630265812525974921, 1.82674360579125943933669988774, 2.03210885974618049944283195420, 2.05554342746935608488702754912, 2.08578209500129275481962360134, 2.18632794480250948746434363449, 2.21812889766920245605260256247, 2.22882784808710970182081812103, 2.23768410938006552614946621067, 2.31737002505866217139330607921, 2.33512230506469217832493818583, 2.36472050741883705781292792626, 2.42304849285641513514818016302
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.