Dirichlet series
| L(s) = 1 | + 2·4-s − 4·5-s − 30·7-s − 6·9-s − 10·11-s + 30·13-s − 4·16-s − 10·17-s − 8·20-s + 132·23-s + 57·25-s − 60·28-s + 160·29-s + 10·31-s + 120·35-s − 12·36-s − 126·37-s − 120·41-s − 20·44-s + 24·45-s − 150·47-s + 457·49-s + 60·52-s + 342·53-s + 40·55-s + 110·59-s − 90·61-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4/5·5-s − 4.28·7-s − 2/3·9-s − 0.909·11-s + 2.30·13-s − 1/4·16-s − 0.588·17-s − 2/5·20-s + 5.73·23-s + 2.27·25-s − 2.14·28-s + 5.51·29-s + 0.322·31-s + 24/7·35-s − 1/3·36-s − 3.40·37-s − 2.92·41-s − 0.454·44-s + 8/15·45-s − 3.19·47-s + 9.32·49-s + 1.15·52-s + 6.45·53-s + 8/11·55-s + 1.86·59-s − 1.47·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.182634\) |
| Root analytic conductor: | \(0.948253\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 11^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.5278312523\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5278312523\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 10 T + 125 T^{2} + 530 T^{3} - 501 T^{4} - 13270 p T^{5} + 11855 p^{2} T^{6} + 18450 p^{3} T^{7} + 25936 p^{4} T^{8} + 18450 p^{5} T^{9} + 11855 p^{6} T^{10} - 13270 p^{7} T^{11} - 501 p^{8} T^{12} + 530 p^{10} T^{13} + 125 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
| good | 2 | \( 1 - p T^{2} + p^{3} T^{4} + p^{2} T^{6} - 95 p T^{7} - 123 T^{8} + 165 p T^{9} - 115 p^{2} T^{10} - 335 p T^{11} + 107 T^{12} - 1635 p T^{13} + 14283 T^{14} + 15195 p T^{15} - 41575 T^{16} + 15195 p^{3} T^{17} + 14283 p^{4} T^{18} - 1635 p^{7} T^{19} + 107 p^{8} T^{20} - 335 p^{11} T^{21} - 115 p^{14} T^{22} + 165 p^{15} T^{23} - 123 p^{16} T^{24} - 95 p^{19} T^{25} + p^{22} T^{26} + p^{27} T^{28} - p^{29} T^{30} + p^{32} T^{32} \) |
| 5 | \( 1 + 4 T - 41 T^{2} - 326 T^{3} + 556 T^{4} + 8402 T^{5} - 24744 T^{6} - 158358 T^{7} + 1206372 T^{8} + 5683018 T^{9} - 36533679 T^{10} - 90990212 T^{11} + 1135973351 T^{12} + 223262516 T^{13} - 39368366016 T^{14} - 2447632564 T^{15} + 1162197815296 T^{16} - 2447632564 p^{2} T^{17} - 39368366016 p^{4} T^{18} + 223262516 p^{6} T^{19} + 1135973351 p^{8} T^{20} - 90990212 p^{10} T^{21} - 36533679 p^{12} T^{22} + 5683018 p^{14} T^{23} + 1206372 p^{16} T^{24} - 158358 p^{18} T^{25} - 24744 p^{20} T^{26} + 8402 p^{22} T^{27} + 556 p^{24} T^{28} - 326 p^{26} T^{29} - 41 p^{28} T^{30} + 4 p^{30} T^{31} + p^{32} T^{32} \) | |
| 7 | \( 1 + 30 T + 443 T^{2} + 5050 T^{3} + 8003 p T^{4} + 586310 T^{5} + 5549342 T^{6} + 50952170 T^{7} + 468313746 T^{8} + 4074185190 T^{9} + 33114120719 T^{10} + 264330080230 T^{11} + 2092010733594 T^{12} + 15864828691680 T^{13} + 116529967249246 T^{14} + 855941511341600 T^{15} + 6146534444128017 T^{16} + 855941511341600 p^{2} T^{17} + 116529967249246 p^{4} T^{18} + 15864828691680 p^{6} T^{19} + 2092010733594 p^{8} T^{20} + 264330080230 p^{10} T^{21} + 33114120719 p^{12} T^{22} + 4074185190 p^{14} T^{23} + 468313746 p^{16} T^{24} + 50952170 p^{18} T^{25} + 5549342 p^{20} T^{26} + 586310 p^{22} T^{27} + 8003 p^{25} T^{28} + 5050 p^{26} T^{29} + 443 p^{28} T^{30} + 30 p^{30} T^{31} + p^{32} T^{32} \) | |
| 13 | \( 1 - 30 T + 587 T^{2} - 11750 T^{3} + 232289 T^{4} - 3942850 T^{5} + 60373070 T^{6} - 849756370 T^{7} + 11683550898 T^{8} - 148143792450 T^{9} + 1704050844755 T^{10} - 17335247746130 T^{11} + 149383368485646 T^{12} - 86339611612200 p T^{13} + 6515222601436258 T^{14} + 31611597343900820 T^{15} - 1160324628036481527 T^{16} + 31611597343900820 p^{2} T^{17} + 6515222601436258 p^{4} T^{18} - 86339611612200 p^{7} T^{19} + 149383368485646 p^{8} T^{20} - 17335247746130 p^{10} T^{21} + 1704050844755 p^{12} T^{22} - 148143792450 p^{14} T^{23} + 11683550898 p^{16} T^{24} - 849756370 p^{18} T^{25} + 60373070 p^{20} T^{26} - 3942850 p^{22} T^{27} + 232289 p^{24} T^{28} - 11750 p^{26} T^{29} + 587 p^{28} T^{30} - 30 p^{30} T^{31} + p^{32} T^{32} \) | |
| 17 | \( 1 + 10 T + 820 T^{2} + 21310 T^{3} + 390646 T^{4} + 13325660 T^{5} + 198026940 T^{6} + 4017398430 T^{7} + 78579088635 T^{8} + 650290141900 T^{9} + 10925791113180 T^{10} + 18747432168760 T^{11} - 4542291295151416 T^{12} - 85648097191686310 T^{13} - 2616538510506970920 T^{14} - 56626351206035192470 T^{15} - \)\(81\!\cdots\!11\)\( T^{16} - 56626351206035192470 p^{2} T^{17} - 2616538510506970920 p^{4} T^{18} - 85648097191686310 p^{6} T^{19} - 4542291295151416 p^{8} T^{20} + 18747432168760 p^{10} T^{21} + 10925791113180 p^{12} T^{22} + 650290141900 p^{14} T^{23} + 78579088635 p^{16} T^{24} + 4017398430 p^{18} T^{25} + 198026940 p^{20} T^{26} + 13325660 p^{22} T^{27} + 390646 p^{24} T^{28} + 21310 p^{26} T^{29} + 820 p^{28} T^{30} + 10 p^{30} T^{31} + p^{32} T^{32} \) | |
| 19 | \( 1 + 68 T^{2} - 24500 T^{3} + 194954 T^{4} - 1666000 T^{5} + 288026060 T^{6} - 3913548340 T^{7} + 46342771203 T^{8} - 120885646560 p T^{9} + 48518556742580 T^{10} - 543301305394100 T^{11} + 17817435718051236 T^{12} - 505288210780128180 T^{13} + 4817696834455760272 T^{14} - \)\(11\!\cdots\!20\)\( T^{15} + \)\(42\!\cdots\!73\)\( T^{16} - \)\(11\!\cdots\!20\)\( p^{2} T^{17} + 4817696834455760272 p^{4} T^{18} - 505288210780128180 p^{6} T^{19} + 17817435718051236 p^{8} T^{20} - 543301305394100 p^{10} T^{21} + 48518556742580 p^{12} T^{22} - 120885646560 p^{15} T^{23} + 46342771203 p^{16} T^{24} - 3913548340 p^{18} T^{25} + 288026060 p^{20} T^{26} - 1666000 p^{22} T^{27} + 194954 p^{24} T^{28} - 24500 p^{26} T^{29} + 68 p^{28} T^{30} + p^{32} T^{32} \) | |
| 23 | \( ( 1 - 66 T + 5094 T^{2} - 230356 T^{3} + 10182141 T^{4} - 348333808 T^{5} + 11089711126 T^{6} - 298981380378 T^{7} + 7385976715852 T^{8} - 298981380378 p^{2} T^{9} + 11089711126 p^{4} T^{10} - 348333808 p^{6} T^{11} + 10182141 p^{8} T^{12} - 230356 p^{10} T^{13} + 5094 p^{12} T^{14} - 66 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 29 | \( 1 - 160 T + 15013 T^{2} - 990140 T^{3} + 49719364 T^{4} - 1931787020 T^{5} + 55026326860 T^{6} - 840259706100 T^{7} - 19139100675292 T^{8} + 2108640129773860 T^{9} - 96767641133217765 T^{10} + 3027695154885413600 T^{11} - 62310681933819745809 T^{12} + \)\(27\!\cdots\!60\)\( T^{13} + \)\(45\!\cdots\!32\)\( T^{14} - \)\(25\!\cdots\!40\)\( T^{15} + \)\(88\!\cdots\!68\)\( T^{16} - \)\(25\!\cdots\!40\)\( p^{2} T^{17} + \)\(45\!\cdots\!32\)\( p^{4} T^{18} + \)\(27\!\cdots\!60\)\( p^{6} T^{19} - 62310681933819745809 p^{8} T^{20} + 3027695154885413600 p^{10} T^{21} - 96767641133217765 p^{12} T^{22} + 2108640129773860 p^{14} T^{23} - 19139100675292 p^{16} T^{24} - 840259706100 p^{18} T^{25} + 55026326860 p^{20} T^{26} - 1931787020 p^{22} T^{27} + 49719364 p^{24} T^{28} - 990140 p^{26} T^{29} + 15013 p^{28} T^{30} - 160 p^{30} T^{31} + p^{32} T^{32} \) | |
| 31 | \( 1 - 10 T - 2495 T^{2} - 50440 T^{3} + 4737748 T^{4} + 140647810 T^{5} - 3464845740 T^{6} - 277335042990 T^{7} + 1019957604708 T^{8} + 9469169297960 p T^{9} + 4654315812995055 T^{10} - 311592266592737110 T^{11} - 7772692308833591089 T^{12} + \)\(19\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} - \)\(10\!\cdots\!00\)\( T^{15} - \)\(12\!\cdots\!00\)\( T^{16} - \)\(10\!\cdots\!00\)\( p^{2} T^{17} + \)\(12\!\cdots\!00\)\( p^{4} T^{18} + \)\(19\!\cdots\!00\)\( p^{6} T^{19} - 7772692308833591089 p^{8} T^{20} - 311592266592737110 p^{10} T^{21} + 4654315812995055 p^{12} T^{22} + 9469169297960 p^{15} T^{23} + 1019957604708 p^{16} T^{24} - 277335042990 p^{18} T^{25} - 3464845740 p^{20} T^{26} + 140647810 p^{22} T^{27} + 4737748 p^{24} T^{28} - 50440 p^{26} T^{29} - 2495 p^{28} T^{30} - 10 p^{30} T^{31} + p^{32} T^{32} \) | |
| 37 | \( 1 + 126 T + 1723 T^{2} - 412814 T^{3} - 18214575 T^{4} + 351923510 T^{5} + 43093576938 T^{6} + 836591530018 T^{7} - 32958031243238 T^{8} - 2093615440151926 T^{9} - 19792102407878669 T^{10} + 1590648037174305038 T^{11} + 62026368402590720226 T^{12} + \)\(49\!\cdots\!72\)\( T^{13} - \)\(40\!\cdots\!62\)\( T^{14} - \)\(78\!\cdots\!60\)\( T^{15} + \)\(21\!\cdots\!13\)\( T^{16} - \)\(78\!\cdots\!60\)\( p^{2} T^{17} - \)\(40\!\cdots\!62\)\( p^{4} T^{18} + \)\(49\!\cdots\!72\)\( p^{6} T^{19} + 62026368402590720226 p^{8} T^{20} + 1590648037174305038 p^{10} T^{21} - 19792102407878669 p^{12} T^{22} - 2093615440151926 p^{14} T^{23} - 32958031243238 p^{16} T^{24} + 836591530018 p^{18} T^{25} + 43093576938 p^{20} T^{26} + 351923510 p^{22} T^{27} - 18214575 p^{24} T^{28} - 412814 p^{26} T^{29} + 1723 p^{28} T^{30} + 126 p^{30} T^{31} + p^{32} T^{32} \) | |
| 41 | \( 1 + 120 T + 11625 T^{2} + 945480 T^{3} + 63040941 T^{4} + 3835690920 T^{5} + 205787619310 T^{6} + 10127082489930 T^{7} + 464765909051670 T^{8} + 19526181575237790 T^{9} + 774755421171593245 T^{10} + 28972958068309133550 T^{11} + \)\(10\!\cdots\!74\)\( T^{12} + \)\(34\!\cdots\!70\)\( T^{13} + \)\(11\!\cdots\!10\)\( T^{14} + \)\(37\!\cdots\!70\)\( T^{15} + \)\(14\!\cdots\!49\)\( T^{16} + \)\(37\!\cdots\!70\)\( p^{2} T^{17} + \)\(11\!\cdots\!10\)\( p^{4} T^{18} + \)\(34\!\cdots\!70\)\( p^{6} T^{19} + \)\(10\!\cdots\!74\)\( p^{8} T^{20} + 28972958068309133550 p^{10} T^{21} + 774755421171593245 p^{12} T^{22} + 19526181575237790 p^{14} T^{23} + 464765909051670 p^{16} T^{24} + 10127082489930 p^{18} T^{25} + 205787619310 p^{20} T^{26} + 3835690920 p^{22} T^{27} + 63040941 p^{24} T^{28} + 945480 p^{26} T^{29} + 11625 p^{28} T^{30} + 120 p^{30} T^{31} + p^{32} T^{32} \) | |
| 43 | \( 1 - 12876 T^{2} + 91501662 T^{4} - 458248399064 T^{6} + 1782172921950441 T^{8} - 5666821055429722520 T^{10} + \)\(15\!\cdots\!78\)\( T^{12} - \)\(34\!\cdots\!80\)\( T^{14} + \)\(69\!\cdots\!12\)\( T^{16} - \)\(34\!\cdots\!80\)\( p^{4} T^{18} + \)\(15\!\cdots\!78\)\( p^{8} T^{20} - 5666821055429722520 p^{12} T^{22} + 1782172921950441 p^{16} T^{24} - 458248399064 p^{20} T^{26} + 91501662 p^{24} T^{28} - 12876 p^{28} T^{30} + p^{32} T^{32} \) | |
| 47 | \( 1 + 150 T + 5703 T^{2} - 111290 T^{3} + 1189236 T^{4} + 1478901940 T^{5} + 65812406892 T^{6} - 349108466940 T^{7} - 11262862832804 T^{8} + 7417285475912590 T^{9} + 389924112150834429 T^{10} + 6636949241471966250 T^{11} + \)\(25\!\cdots\!19\)\( T^{12} + \)\(23\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!16\)\( T^{14} + \)\(44\!\cdots\!40\)\( T^{15} + \)\(22\!\cdots\!12\)\( T^{16} + \)\(44\!\cdots\!40\)\( p^{2} T^{17} + \)\(10\!\cdots\!16\)\( p^{4} T^{18} + \)\(23\!\cdots\!60\)\( p^{6} T^{19} + \)\(25\!\cdots\!19\)\( p^{8} T^{20} + 6636949241471966250 p^{10} T^{21} + 389924112150834429 p^{12} T^{22} + 7417285475912590 p^{14} T^{23} - 11262862832804 p^{16} T^{24} - 349108466940 p^{18} T^{25} + 65812406892 p^{20} T^{26} + 1478901940 p^{22} T^{27} + 1189236 p^{24} T^{28} - 111290 p^{26} T^{29} + 5703 p^{28} T^{30} + 150 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 - 342 T + 56425 T^{2} - 5825308 T^{3} + 395226372 T^{4} - 14794732502 T^{5} - 224283930792 T^{6} + 78367975406558 T^{7} - 6103771119756596 T^{8} + 258849959496378940 T^{9} - 2857462886249188349 T^{10} - \)\(46\!\cdots\!66\)\( T^{11} + \)\(40\!\cdots\!59\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!16\)\( T^{14} + \)\(15\!\cdots\!92\)\( T^{15} - \)\(14\!\cdots\!56\)\( T^{16} + \)\(15\!\cdots\!92\)\( p^{2} T^{17} + \)\(30\!\cdots\!16\)\( p^{4} T^{18} - \)\(17\!\cdots\!00\)\( p^{6} T^{19} + \)\(40\!\cdots\!59\)\( p^{8} T^{20} - \)\(46\!\cdots\!66\)\( p^{10} T^{21} - 2857462886249188349 p^{12} T^{22} + 258849959496378940 p^{14} T^{23} - 6103771119756596 p^{16} T^{24} + 78367975406558 p^{18} T^{25} - 224283930792 p^{20} T^{26} - 14794732502 p^{22} T^{27} + 395226372 p^{24} T^{28} - 5825308 p^{26} T^{29} + 56425 p^{28} T^{30} - 342 p^{30} T^{31} + p^{32} T^{32} \) | |
| 59 | \( 1 - 110 T - 1820 T^{2} + 874210 T^{3} - 21932762 T^{4} - 3787520260 T^{5} + 169192636440 T^{6} + 16984414752090 T^{7} - 1089925211772957 T^{8} - 63720166969996940 T^{9} + 5917794901333836780 T^{10} + \)\(16\!\cdots\!40\)\( T^{11} - \)\(24\!\cdots\!44\)\( T^{12} - \)\(47\!\cdots\!50\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!50\)\( T^{15} - \)\(39\!\cdots\!75\)\( T^{16} + \)\(83\!\cdots\!50\)\( p^{2} T^{17} + \)\(10\!\cdots\!00\)\( p^{4} T^{18} - \)\(47\!\cdots\!50\)\( p^{6} T^{19} - \)\(24\!\cdots\!44\)\( p^{8} T^{20} + \)\(16\!\cdots\!40\)\( p^{10} T^{21} + 5917794901333836780 p^{12} T^{22} - 63720166969996940 p^{14} T^{23} - 1089925211772957 p^{16} T^{24} + 16984414752090 p^{18} T^{25} + 169192636440 p^{20} T^{26} - 3787520260 p^{22} T^{27} - 21932762 p^{24} T^{28} + 874210 p^{26} T^{29} - 1820 p^{28} T^{30} - 110 p^{30} T^{31} + p^{32} T^{32} \) | |
| 61 | \( 1 + 90 T + 85 p T^{2} + 111300 T^{3} + 5561708 T^{4} + 355418970 T^{5} + 44628720040 T^{6} + 1586771608090 T^{7} - 2104031885352 T^{8} - 10404552243962580 T^{9} - 1081281936652696465 T^{10} - 65423743873501326530 T^{11} - \)\(19\!\cdots\!49\)\( T^{12} + \)\(44\!\cdots\!80\)\( T^{13} - \)\(23\!\cdots\!00\)\( T^{14} - \)\(53\!\cdots\!80\)\( T^{15} - \)\(50\!\cdots\!80\)\( T^{16} - \)\(53\!\cdots\!80\)\( p^{2} T^{17} - \)\(23\!\cdots\!00\)\( p^{4} T^{18} + \)\(44\!\cdots\!80\)\( p^{6} T^{19} - \)\(19\!\cdots\!49\)\( p^{8} T^{20} - 65423743873501326530 p^{10} T^{21} - 1081281936652696465 p^{12} T^{22} - 10404552243962580 p^{14} T^{23} - 2104031885352 p^{16} T^{24} + 1586771608090 p^{18} T^{25} + 44628720040 p^{20} T^{26} + 355418970 p^{22} T^{27} + 5561708 p^{24} T^{28} + 111300 p^{26} T^{29} + 85 p^{29} T^{30} + 90 p^{30} T^{31} + p^{32} T^{32} \) | |
| 67 | \( ( 1 - 18 T + 21975 T^{2} - 668180 T^{3} + 239231415 T^{4} - 8253401324 T^{5} + 1771382320897 T^{6} - 55179990303630 T^{7} + 9425465643780880 T^{8} - 55179990303630 p^{2} T^{9} + 1771382320897 p^{4} T^{10} - 8253401324 p^{6} T^{11} + 239231415 p^{8} T^{12} - 668180 p^{10} T^{13} + 21975 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 + 236 T + 10027 T^{2} - 2371506 T^{3} - 303058608 T^{4} - 188446842 T^{5} + 2379677829156 T^{6} + 145240013200230 T^{7} - 6446716975029360 T^{8} - 1111666934803343322 T^{9} - 20084120671257805311 T^{10} + \)\(40\!\cdots\!44\)\( T^{11} + \)\(23\!\cdots\!83\)\( T^{12} - \)\(82\!\cdots\!76\)\( T^{13} - \)\(11\!\cdots\!12\)\( T^{14} + \)\(77\!\cdots\!36\)\( T^{15} + \)\(49\!\cdots\!64\)\( T^{16} + \)\(77\!\cdots\!36\)\( p^{2} T^{17} - \)\(11\!\cdots\!12\)\( p^{4} T^{18} - \)\(82\!\cdots\!76\)\( p^{6} T^{19} + \)\(23\!\cdots\!83\)\( p^{8} T^{20} + \)\(40\!\cdots\!44\)\( p^{10} T^{21} - 20084120671257805311 p^{12} T^{22} - 1111666934803343322 p^{14} T^{23} - 6446716975029360 p^{16} T^{24} + 145240013200230 p^{18} T^{25} + 2379677829156 p^{20} T^{26} - 188446842 p^{22} T^{27} - 303058608 p^{24} T^{28} - 2371506 p^{26} T^{29} + 10027 p^{28} T^{30} + 236 p^{30} T^{31} + p^{32} T^{32} \) | |
| 73 | \( 1 + 350 T + 71113 T^{2} + 10598660 T^{3} + 1285017928 T^{4} + 133684739230 T^{5} + 12256794539364 T^{6} + 1004390235106950 T^{7} + 73846073626981752 T^{8} + 4829859050716130540 T^{9} + \)\(26\!\cdots\!55\)\( T^{10} + \)\(10\!\cdots\!50\)\( T^{11} + \)\(86\!\cdots\!87\)\( T^{12} - \)\(60\!\cdots\!80\)\( T^{13} - \)\(85\!\cdots\!12\)\( T^{14} - \)\(83\!\cdots\!20\)\( T^{15} - \)\(66\!\cdots\!40\)\( T^{16} - \)\(83\!\cdots\!20\)\( p^{2} T^{17} - \)\(85\!\cdots\!12\)\( p^{4} T^{18} - \)\(60\!\cdots\!80\)\( p^{6} T^{19} + \)\(86\!\cdots\!87\)\( p^{8} T^{20} + \)\(10\!\cdots\!50\)\( p^{10} T^{21} + \)\(26\!\cdots\!55\)\( p^{12} T^{22} + 4829859050716130540 p^{14} T^{23} + 73846073626981752 p^{16} T^{24} + 1004390235106950 p^{18} T^{25} + 12256794539364 p^{20} T^{26} + 133684739230 p^{22} T^{27} + 1285017928 p^{24} T^{28} + 10598660 p^{26} T^{29} + 71113 p^{28} T^{30} + 350 p^{30} T^{31} + p^{32} T^{32} \) | |
| 79 | \( 1 - 210 T + 33391 T^{2} - 5683230 T^{3} + 793281737 T^{4} - 99138120690 T^{5} + 12095814452554 T^{6} - 1372834116707750 T^{7} + 146848726194339006 T^{8} - 15133095124360319250 T^{9} + \)\(14\!\cdots\!75\)\( T^{10} - \)\(14\!\cdots\!50\)\( T^{11} + \)\(12\!\cdots\!18\)\( T^{12} - \)\(11\!\cdots\!40\)\( T^{13} + \)\(98\!\cdots\!50\)\( T^{14} - \)\(81\!\cdots\!40\)\( T^{15} + \)\(65\!\cdots\!97\)\( T^{16} - \)\(81\!\cdots\!40\)\( p^{2} T^{17} + \)\(98\!\cdots\!50\)\( p^{4} T^{18} - \)\(11\!\cdots\!40\)\( p^{6} T^{19} + \)\(12\!\cdots\!18\)\( p^{8} T^{20} - \)\(14\!\cdots\!50\)\( p^{10} T^{21} + \)\(14\!\cdots\!75\)\( p^{12} T^{22} - 15133095124360319250 p^{14} T^{23} + 146848726194339006 p^{16} T^{24} - 1372834116707750 p^{18} T^{25} + 12095814452554 p^{20} T^{26} - 99138120690 p^{22} T^{27} + 793281737 p^{24} T^{28} - 5683230 p^{26} T^{29} + 33391 p^{28} T^{30} - 210 p^{30} T^{31} + p^{32} T^{32} \) | |
| 83 | \( 1 + 190 T + 67177 T^{2} + 11048590 T^{3} + 2116773229 T^{4} + 299366635430 T^{5} + 41314892504790 T^{6} + 4967390563524780 T^{7} + 550864440609675318 T^{8} + 55772626453864581280 T^{9} + \)\(52\!\cdots\!25\)\( T^{10} + \)\(44\!\cdots\!60\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} + \)\(26\!\cdots\!50\)\( T^{13} + \)\(18\!\cdots\!58\)\( T^{14} + \)\(13\!\cdots\!30\)\( T^{15} + \)\(10\!\cdots\!33\)\( T^{16} + \)\(13\!\cdots\!30\)\( p^{2} T^{17} + \)\(18\!\cdots\!58\)\( p^{4} T^{18} + \)\(26\!\cdots\!50\)\( p^{6} T^{19} + \)\(35\!\cdots\!06\)\( p^{8} T^{20} + \)\(44\!\cdots\!60\)\( p^{10} T^{21} + \)\(52\!\cdots\!25\)\( p^{12} T^{22} + 55772626453864581280 p^{14} T^{23} + 550864440609675318 p^{16} T^{24} + 4967390563524780 p^{18} T^{25} + 41314892504790 p^{20} T^{26} + 299366635430 p^{22} T^{27} + 2116773229 p^{24} T^{28} + 11048590 p^{26} T^{29} + 67177 p^{28} T^{30} + 190 p^{30} T^{31} + p^{32} T^{32} \) | |
| 89 | \( ( 1 - 38 T + 37118 T^{2} - 1566244 T^{3} + 705775677 T^{4} - 28974031040 T^{5} + 8919242373542 T^{6} - 334883278291638 T^{7} + 81740346306794380 T^{8} - 334883278291638 p^{2} T^{9} + 8919242373542 p^{4} T^{10} - 28974031040 p^{6} T^{11} + 705775677 p^{8} T^{12} - 1566244 p^{10} T^{13} + 37118 p^{12} T^{14} - 38 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( 1 + 354 T + 19173 T^{2} - 6545856 T^{3} - 673044900 T^{4} + 77524200990 T^{5} + 12627807964588 T^{6} - 188800913648178 T^{7} - 100847199348278148 T^{8} - 1111689883210016304 T^{9} + \)\(55\!\cdots\!71\)\( T^{10} - \)\(94\!\cdots\!38\)\( T^{11} - \)\(58\!\cdots\!69\)\( T^{12} + \)\(47\!\cdots\!48\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} - \)\(16\!\cdots\!52\)\( T^{16} - \)\(29\!\cdots\!40\)\( p^{2} T^{17} + \)\(12\!\cdots\!68\)\( p^{4} T^{18} + \)\(47\!\cdots\!48\)\( p^{6} T^{19} - \)\(58\!\cdots\!69\)\( p^{8} T^{20} - \)\(94\!\cdots\!38\)\( p^{10} T^{21} + \)\(55\!\cdots\!71\)\( p^{12} T^{22} - 1111689883210016304 p^{14} T^{23} - 100847199348278148 p^{16} T^{24} - 188800913648178 p^{18} T^{25} + 12627807964588 p^{20} T^{26} + 77524200990 p^{22} T^{27} - 673044900 p^{24} T^{28} - 6545856 p^{26} T^{29} + 19173 p^{28} T^{30} + 354 p^{30} T^{31} + p^{32} T^{32} \) | |
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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
−5.49834891209103864267408730009, −5.29323744001235587798667716057, −5.20797058097258475114000588228, −5.14317502831064095892490996680, −5.11956771808791459996793652974, −4.82791462362337011256139196656, −4.63787193544416285292111844226, −4.54428452289288398947467468480, −4.41321585574761399572764056387, −4.38862001674889299598542538852, −4.03731350739414275373144785718, −4.00793211295959677431621416426, −3.69790791199141113395245674101, −3.49922187538414522079252699550, −3.41865471932383632168954996753, −3.21440352003552587764182549672, −3.17871944348555400207830985126, −2.97880469580794270112126191909, −2.85141994790067530153574085742, −2.80288377966965723474973235437, −2.75245676862392108387586191427, −2.67195670271665527932868624657, −1.84109453060867794225412917651, −1.28812877788889838587662610745, −0.884389385998805316190511189226, 0.884389385998805316190511189226, 1.28812877788889838587662610745, 1.84109453060867794225412917651, 2.67195670271665527932868624657, 2.75245676862392108387586191427, 2.80288377966965723474973235437, 2.85141994790067530153574085742, 2.97880469580794270112126191909, 3.17871944348555400207830985126, 3.21440352003552587764182549672, 3.41865471932383632168954996753, 3.49922187538414522079252699550, 3.69790791199141113395245674101, 4.00793211295959677431621416426, 4.03731350739414275373144785718, 4.38862001674889299598542538852, 4.41321585574761399572764056387, 4.54428452289288398947467468480, 4.63787193544416285292111844226, 4.82791462362337011256139196656, 5.11956771808791459996793652974, 5.14317502831064095892490996680, 5.20797058097258475114000588228, 5.29323744001235587798667716057, 5.49834891209103864267408730009
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.