Dirichlet series
| L(s) = 1 | + 28·2-s + 392·4-s + 36·5-s − 48·7-s + 3.58e3·8-s + 394·9-s + 1.00e3·10-s − 104·13-s − 1.34e3·14-s + 2.32e4·16-s + 516·17-s + 1.10e4·18-s − 328·19-s + 1.41e4·20-s + 154·23-s + 648·25-s − 2.91e3·26-s − 1.88e4·28-s + 1.68e3·29-s + 3.83e3·31-s + 1.07e5·32-s + 1.44e4·34-s − 1.72e3·35-s + 1.54e5·36-s + 2.64e3·37-s − 9.18e3·38-s + 1.29e5·40-s + ⋯ |
| L(s) = 1 | + 7·2-s + 49/2·4-s + 1.43·5-s − 0.979·7-s + 56·8-s + 4.86·9-s + 10.0·10-s − 0.615·13-s − 6.85·14-s + 91·16-s + 1.78·17-s + 34.0·18-s − 0.908·19-s + 35.2·20-s + 0.291·23-s + 1.03·25-s − 4.30·26-s − 24·28-s + 2.00·29-s + 3.98·31-s + 105·32-s + 12.4·34-s − 1.41·35-s + 119.·36-s + 1.92·37-s − 6.36·38-s + 80.6·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.34836\times 10^{12}\) |
| Root analytic conductor: | \(2.76575\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [2]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(2919.496469\) |
| \(L(\frac12)\) | \(\approx\) | \(2919.496469\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{2} T + p^{3} T^{2} )^{7} \) |
| 37 | \( 1 - 2640 T + 405935 T^{2} + 965104544 T^{3} + 8034076601 p^{2} T^{4} - 16765958301104 p^{2} T^{5} + 6740332998543 p^{4} T^{6} + 528102092992 p^{6} T^{7} + 6740332998543 p^{8} T^{8} - 16765958301104 p^{10} T^{9} + 8034076601 p^{14} T^{10} + 965104544 p^{16} T^{11} + 405935 p^{20} T^{12} - 2640 p^{24} T^{13} + p^{28} T^{14} \) | |
| good | 3 | \( 1 - 394 T^{2} + 93115 T^{4} - 587366 p^{3} T^{6} + 80604635 p^{3} T^{8} - 3097675370 p^{4} T^{10} + 34293047684 p^{6} T^{12} - 330117299942 p^{8} T^{14} + 34293047684 p^{14} T^{16} - 3097675370 p^{20} T^{18} + 80604635 p^{27} T^{20} - 587366 p^{35} T^{22} + 93115 p^{40} T^{24} - 394 p^{48} T^{26} + p^{56} T^{28} \) |
| 5 | \( 1 - 36 T + 648 T^{2} + 31932 T^{3} - 438986 T^{4} - 14206344 T^{5} + 1305717624 T^{6} - 3982435044 T^{7} - 508187230551 T^{8} + 19132097149128 T^{9} + 187484574015864 T^{10} - 10770262150069536 T^{11} + 217306499980176416 T^{12} + 4704213655911751596 T^{13} - \)\(17\!\cdots\!12\)\( T^{14} + 4704213655911751596 p^{4} T^{15} + 217306499980176416 p^{8} T^{16} - 10770262150069536 p^{12} T^{17} + 187484574015864 p^{16} T^{18} + 19132097149128 p^{20} T^{19} - 508187230551 p^{24} T^{20} - 3982435044 p^{28} T^{21} + 1305717624 p^{32} T^{22} - 14206344 p^{36} T^{23} - 438986 p^{40} T^{24} + 31932 p^{44} T^{25} + 648 p^{48} T^{26} - 36 p^{52} T^{27} + p^{56} T^{28} \) | |
| 7 | \( ( 1 + 24 T + 5478 T^{2} + 47678 p T^{3} + 26727930 T^{4} + 1288490732 T^{5} + 12995464045 p T^{6} + 3959986793148 T^{7} + 12995464045 p^{5} T^{8} + 1288490732 p^{8} T^{9} + 26727930 p^{12} T^{10} + 47678 p^{17} T^{11} + 5478 p^{20} T^{12} + 24 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 11 | \( 1 - 80778 T^{2} + 2466769755 T^{4} - 27246453626146 T^{6} - 185794956730366311 T^{8} + \)\(41\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(37\!\cdots\!06\)\( T^{14} + \)\(12\!\cdots\!76\)\( p^{8} T^{16} + \)\(41\!\cdots\!06\)\( p^{16} T^{18} - 185794956730366311 p^{24} T^{20} - 27246453626146 p^{32} T^{22} + 2466769755 p^{40} T^{24} - 80778 p^{48} T^{26} + p^{56} T^{28} \) | |
| 13 | \( 1 + 8 p T + 32 p^{2} T^{2} - 9277480 T^{3} - 2208851818 T^{4} + 19613930836 T^{5} + 57021137013888 T^{6} + 1307680498718384 p T^{7} + 1387241892030126793 T^{8} - \)\(32\!\cdots\!68\)\( T^{9} - \)\(73\!\cdots\!44\)\( T^{10} - \)\(12\!\cdots\!32\)\( T^{11} - \)\(20\!\cdots\!92\)\( T^{12} + \)\(34\!\cdots\!60\)\( T^{13} + \)\(51\!\cdots\!44\)\( T^{14} + \)\(34\!\cdots\!60\)\( p^{4} T^{15} - \)\(20\!\cdots\!92\)\( p^{8} T^{16} - \)\(12\!\cdots\!32\)\( p^{12} T^{17} - \)\(73\!\cdots\!44\)\( p^{16} T^{18} - \)\(32\!\cdots\!68\)\( p^{20} T^{19} + 1387241892030126793 p^{24} T^{20} + 1307680498718384 p^{29} T^{21} + 57021137013888 p^{32} T^{22} + 19613930836 p^{36} T^{23} - 2208851818 p^{40} T^{24} - 9277480 p^{44} T^{25} + 32 p^{50} T^{26} + 8 p^{53} T^{27} + p^{56} T^{28} \) | |
| 17 | \( 1 - 516 T + 133128 T^{2} - 47886004 T^{3} + 41221820039 T^{4} - 16261047800120 T^{5} + 4049456896253936 T^{6} - 1386603073040432344 T^{7} + 41495294402678657189 p T^{8} - \)\(23\!\cdots\!24\)\( T^{9} + \)\(57\!\cdots\!92\)\( T^{10} - \)\(19\!\cdots\!76\)\( T^{11} + \)\(72\!\cdots\!43\)\( T^{12} - \)\(20\!\cdots\!28\)\( T^{13} + \)\(49\!\cdots\!40\)\( T^{14} - \)\(20\!\cdots\!28\)\( p^{4} T^{15} + \)\(72\!\cdots\!43\)\( p^{8} T^{16} - \)\(19\!\cdots\!76\)\( p^{12} T^{17} + \)\(57\!\cdots\!92\)\( p^{16} T^{18} - \)\(23\!\cdots\!24\)\( p^{20} T^{19} + 41495294402678657189 p^{25} T^{20} - 1386603073040432344 p^{28} T^{21} + 4049456896253936 p^{32} T^{22} - 16261047800120 p^{36} T^{23} + 41221820039 p^{40} T^{24} - 47886004 p^{44} T^{25} + 133128 p^{48} T^{26} - 516 p^{52} T^{27} + p^{56} T^{28} \) | |
| 19 | \( 1 + 328 T + 53792 T^{2} - 46768048 T^{3} - 61988870857 T^{4} - 17705036729768 T^{5} - 1379121549359008 T^{6} + 2604594823528371000 T^{7} + \)\(14\!\cdots\!73\)\( T^{8} + \)\(18\!\cdots\!44\)\( p T^{9} - \)\(10\!\cdots\!52\)\( T^{10} - \)\(42\!\cdots\!84\)\( T^{11} - \)\(11\!\cdots\!01\)\( T^{12} - \)\(99\!\cdots\!36\)\( T^{13} + \)\(65\!\cdots\!88\)\( T^{14} - \)\(99\!\cdots\!36\)\( p^{4} T^{15} - \)\(11\!\cdots\!01\)\( p^{8} T^{16} - \)\(42\!\cdots\!84\)\( p^{12} T^{17} - \)\(10\!\cdots\!52\)\( p^{16} T^{18} + \)\(18\!\cdots\!44\)\( p^{21} T^{19} + \)\(14\!\cdots\!73\)\( p^{24} T^{20} + 2604594823528371000 p^{28} T^{21} - 1379121549359008 p^{32} T^{22} - 17705036729768 p^{36} T^{23} - 61988870857 p^{40} T^{24} - 46768048 p^{44} T^{25} + 53792 p^{48} T^{26} + 328 p^{52} T^{27} + p^{56} T^{28} \) | |
| 23 | \( 1 - 154 T + 11858 T^{2} + 140031102 T^{3} + 98456010934 T^{4} - 35233787322698 T^{5} + 14062866633707322 T^{6} + 19039999676659854230 T^{7} - 97473444742795096815 T^{8} - \)\(48\!\cdots\!36\)\( T^{9} + \)\(19\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!84\)\( T^{11} - \)\(29\!\cdots\!04\)\( T^{12} + \)\(29\!\cdots\!80\)\( T^{13} + \)\(14\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!80\)\( p^{4} T^{15} - \)\(29\!\cdots\!04\)\( p^{8} T^{16} + \)\(13\!\cdots\!84\)\( p^{12} T^{17} + \)\(19\!\cdots\!28\)\( p^{16} T^{18} - \)\(48\!\cdots\!36\)\( p^{20} T^{19} - 97473444742795096815 p^{24} T^{20} + 19039999676659854230 p^{28} T^{21} + 14062866633707322 p^{32} T^{22} - 35233787322698 p^{36} T^{23} + 98456010934 p^{40} T^{24} + 140031102 p^{44} T^{25} + 11858 p^{48} T^{26} - 154 p^{52} T^{27} + p^{56} T^{28} \) | |
| 29 | \( 1 - 1686 T + 1421298 T^{2} - 1971846790 T^{3} + 1965616110014 T^{4} - 1575963291476618 T^{5} + 1807437745115551826 T^{6} - \)\(15\!\cdots\!10\)\( T^{7} + \)\(11\!\cdots\!85\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} - \)\(81\!\cdots\!60\)\( T^{11} + \)\(83\!\cdots\!76\)\( T^{12} - \)\(73\!\cdots\!96\)\( T^{13} + \)\(52\!\cdots\!24\)\( T^{14} - \)\(73\!\cdots\!96\)\( p^{4} T^{15} + \)\(83\!\cdots\!76\)\( p^{8} T^{16} - \)\(81\!\cdots\!60\)\( p^{12} T^{17} + \)\(10\!\cdots\!40\)\( p^{16} T^{18} - \)\(11\!\cdots\!52\)\( p^{20} T^{19} + \)\(11\!\cdots\!85\)\( p^{24} T^{20} - \)\(15\!\cdots\!10\)\( p^{28} T^{21} + 1807437745115551826 p^{32} T^{22} - 1575963291476618 p^{36} T^{23} + 1965616110014 p^{40} T^{24} - 1971846790 p^{44} T^{25} + 1421298 p^{48} T^{26} - 1686 p^{52} T^{27} + p^{56} T^{28} \) | |
| 31 | \( 1 - 3834 T + 7349778 T^{2} - 11827918610 T^{3} + 15944162032282 T^{4} - 15552397812022890 T^{5} + 12391671200386392914 T^{6} - \)\(78\!\cdots\!94\)\( T^{7} + \)\(12\!\cdots\!29\)\( T^{8} + \)\(20\!\cdots\!08\)\( T^{9} - \)\(11\!\cdots\!92\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(60\!\cdots\!24\)\( T^{12} - \)\(99\!\cdots\!32\)\( T^{13} + \)\(10\!\cdots\!52\)\( T^{14} - \)\(99\!\cdots\!32\)\( p^{4} T^{15} + \)\(60\!\cdots\!24\)\( p^{8} T^{16} - \)\(13\!\cdots\!00\)\( p^{12} T^{17} - \)\(11\!\cdots\!92\)\( p^{16} T^{18} + \)\(20\!\cdots\!08\)\( p^{20} T^{19} + \)\(12\!\cdots\!29\)\( p^{24} T^{20} - \)\(78\!\cdots\!94\)\( p^{28} T^{21} + 12391671200386392914 p^{32} T^{22} - 15552397812022890 p^{36} T^{23} + 15944162032282 p^{40} T^{24} - 11827918610 p^{44} T^{25} + 7349778 p^{48} T^{26} - 3834 p^{52} T^{27} + p^{56} T^{28} \) | |
| 41 | \( 1 - 14748206 T^{2} + 106766935884779 T^{4} - \)\(47\!\cdots\!98\)\( T^{6} + \)\(13\!\cdots\!33\)\( T^{8} - \)\(17\!\cdots\!42\)\( T^{10} - \)\(28\!\cdots\!68\)\( T^{12} + \)\(18\!\cdots\!82\)\( T^{14} - \)\(28\!\cdots\!68\)\( p^{8} T^{16} - \)\(17\!\cdots\!42\)\( p^{16} T^{18} + \)\(13\!\cdots\!33\)\( p^{24} T^{20} - \)\(47\!\cdots\!98\)\( p^{32} T^{22} + 106766935884779 p^{40} T^{24} - 14748206 p^{48} T^{26} + p^{56} T^{28} \) | |
| 43 | \( 1 - 3616 T + 6537728 T^{2} - 6836503144 T^{3} + 714676665701 p T^{4} - 118765218497983432 T^{5} + \)\(25\!\cdots\!76\)\( T^{6} - \)\(34\!\cdots\!96\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(37\!\cdots\!20\)\( T^{10} - \)\(59\!\cdots\!32\)\( T^{11} + \)\(88\!\cdots\!31\)\( T^{12} - \)\(17\!\cdots\!52\)\( T^{13} + \)\(39\!\cdots\!04\)\( T^{14} - \)\(17\!\cdots\!52\)\( p^{4} T^{15} + \)\(88\!\cdots\!31\)\( p^{8} T^{16} - \)\(59\!\cdots\!32\)\( p^{12} T^{17} + \)\(37\!\cdots\!20\)\( p^{16} T^{18} - \)\(15\!\cdots\!80\)\( p^{20} T^{19} + \)\(54\!\cdots\!01\)\( p^{24} T^{20} - \)\(34\!\cdots\!96\)\( p^{28} T^{21} + \)\(25\!\cdots\!76\)\( p^{32} T^{22} - 118765218497983432 p^{36} T^{23} + 714676665701 p^{41} T^{24} - 6836503144 p^{44} T^{25} + 6537728 p^{48} T^{26} - 3616 p^{52} T^{27} + p^{56} T^{28} \) | |
| 47 | \( ( 1 + 3446 T + 27752138 T^{2} + 78844011930 T^{3} + 354673846980296 T^{4} + 827649559852338258 T^{5} + \)\(27\!\cdots\!01\)\( T^{6} + \)\(51\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!01\)\( p^{4} T^{8} + 827649559852338258 p^{8} T^{9} + 354673846980296 p^{12} T^{10} + 78844011930 p^{16} T^{11} + 27752138 p^{20} T^{12} + 3446 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 53 | \( ( 1 - 6286 T + 40676124 T^{2} - 172711547050 T^{3} + 675237931947404 T^{4} - 2184575263774962814 T^{5} + \)\(67\!\cdots\!95\)\( T^{6} - \)\(19\!\cdots\!48\)\( T^{7} + \)\(67\!\cdots\!95\)\( p^{4} T^{8} - 2184575263774962814 p^{8} T^{9} + 675237931947404 p^{12} T^{10} - 172711547050 p^{16} T^{11} + 40676124 p^{20} T^{12} - 6286 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 59 | \( 1 + 8422 T + 35465042 T^{2} + 135503936534 T^{3} + 13910861675085 p T^{4} + 4198129317081506500 T^{5} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(50\!\cdots\!76\)\( T^{7} + \)\(20\!\cdots\!25\)\( T^{8} + \)\(87\!\cdots\!86\)\( T^{9} + \)\(31\!\cdots\!06\)\( T^{10} + \)\(92\!\cdots\!98\)\( T^{11} + \)\(28\!\cdots\!03\)\( T^{12} + \)\(11\!\cdots\!56\)\( T^{13} + \)\(45\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!56\)\( p^{4} T^{15} + \)\(28\!\cdots\!03\)\( p^{8} T^{16} + \)\(92\!\cdots\!98\)\( p^{12} T^{17} + \)\(31\!\cdots\!06\)\( p^{16} T^{18} + \)\(87\!\cdots\!86\)\( p^{20} T^{19} + \)\(20\!\cdots\!25\)\( p^{24} T^{20} + \)\(50\!\cdots\!76\)\( p^{28} T^{21} + \)\(15\!\cdots\!48\)\( p^{32} T^{22} + 4198129317081506500 p^{36} T^{23} + 13910861675085 p^{41} T^{24} + 135503936534 p^{44} T^{25} + 35465042 p^{48} T^{26} + 8422 p^{52} T^{27} + p^{56} T^{28} \) | |
| 61 | \( 1 + 6386 T + 20390498 T^{2} + 113469443482 T^{3} + 752895922791498 T^{4} + 2371715991359512926 T^{5} + \)\(62\!\cdots\!94\)\( T^{6} + \)\(28\!\cdots\!86\)\( T^{7} + \)\(57\!\cdots\!93\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{9} - \)\(55\!\cdots\!12\)\( T^{10} - \)\(40\!\cdots\!96\)\( T^{11} - \)\(37\!\cdots\!56\)\( T^{12} - \)\(13\!\cdots\!40\)\( T^{13} - \)\(37\!\cdots\!08\)\( T^{14} - \)\(13\!\cdots\!40\)\( p^{4} T^{15} - \)\(37\!\cdots\!56\)\( p^{8} T^{16} - \)\(40\!\cdots\!96\)\( p^{12} T^{17} - \)\(55\!\cdots\!12\)\( p^{16} T^{18} - \)\(11\!\cdots\!16\)\( p^{20} T^{19} + \)\(57\!\cdots\!93\)\( p^{24} T^{20} + \)\(28\!\cdots\!86\)\( p^{28} T^{21} + \)\(62\!\cdots\!94\)\( p^{32} T^{22} + 2371715991359512926 p^{36} T^{23} + 752895922791498 p^{40} T^{24} + 113469443482 p^{44} T^{25} + 20390498 p^{48} T^{26} + 6386 p^{52} T^{27} + p^{56} T^{28} \) | |
| 67 | \( 1 - 211225400 T^{2} + 21551595701200454 T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(66\!\cdots\!33\)\( T^{8} - \)\(23\!\cdots\!90\)\( T^{10} + \)\(66\!\cdots\!80\)\( T^{12} - \)\(14\!\cdots\!52\)\( T^{14} + \)\(66\!\cdots\!80\)\( p^{8} T^{16} - \)\(23\!\cdots\!90\)\( p^{16} T^{18} + \)\(66\!\cdots\!33\)\( p^{24} T^{20} - \)\(14\!\cdots\!12\)\( p^{32} T^{22} + 21551595701200454 p^{40} T^{24} - 211225400 p^{48} T^{26} + p^{56} T^{28} \) | |
| 71 | \( ( 1 - 4340 T + 104964556 T^{2} - 461979062294 T^{3} + 5871482815990354 T^{4} - 24337100176138065548 T^{5} + \)\(21\!\cdots\!77\)\( T^{6} - \)\(77\!\cdots\!32\)\( T^{7} + \)\(21\!\cdots\!77\)\( p^{4} T^{8} - 24337100176138065548 p^{8} T^{9} + 5871482815990354 p^{12} T^{10} - 461979062294 p^{16} T^{11} + 104964556 p^{20} T^{12} - 4340 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 73 | \( 1 - 210958246 T^{2} + 20767176635121667 T^{4} - \)\(12\!\cdots\!54\)\( T^{6} + \)\(56\!\cdots\!49\)\( T^{8} - \)\(19\!\cdots\!58\)\( T^{10} + \)\(59\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!94\)\( T^{14} + \)\(59\!\cdots\!28\)\( p^{8} T^{16} - \)\(19\!\cdots\!58\)\( p^{16} T^{18} + \)\(56\!\cdots\!49\)\( p^{24} T^{20} - \)\(12\!\cdots\!54\)\( p^{32} T^{22} + 20767176635121667 p^{40} T^{24} - 210958246 p^{48} T^{26} + p^{56} T^{28} \) | |
| 79 | \( 1 + 28520 T + 406695200 T^{2} + 4177298298752 T^{3} + 34402110843515758 T^{4} + \)\(23\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} + \)\(77\!\cdots\!36\)\( T^{7} + \)\(36\!\cdots\!41\)\( T^{8} + \)\(86\!\cdots\!00\)\( T^{9} - \)\(67\!\cdots\!12\)\( T^{10} - \)\(12\!\cdots\!32\)\( T^{11} - \)\(12\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!12\)\( T^{13} - \)\(61\!\cdots\!52\)\( T^{14} - \)\(93\!\cdots\!12\)\( p^{4} T^{15} - \)\(12\!\cdots\!16\)\( p^{8} T^{16} - \)\(12\!\cdots\!32\)\( p^{12} T^{17} - \)\(67\!\cdots\!12\)\( p^{16} T^{18} + \)\(86\!\cdots\!00\)\( p^{20} T^{19} + \)\(36\!\cdots\!41\)\( p^{24} T^{20} + \)\(77\!\cdots\!36\)\( p^{28} T^{21} + \)\(13\!\cdots\!12\)\( p^{32} T^{22} + \)\(23\!\cdots\!68\)\( p^{36} T^{23} + 34402110843515758 p^{40} T^{24} + 4177298298752 p^{44} T^{25} + 406695200 p^{48} T^{26} + 28520 p^{52} T^{27} + p^{56} T^{28} \) | |
| 83 | \( ( 1 + 11344 T + 233305356 T^{2} + 2018491487626 T^{3} + 25472496965790218 T^{4} + \)\(17\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!09\)\( T^{6} + \)\(99\!\cdots\!68\)\( T^{7} + \)\(17\!\cdots\!09\)\( p^{4} T^{8} + \)\(17\!\cdots\!92\)\( p^{8} T^{9} + 25472496965790218 p^{12} T^{10} + 2018491487626 p^{16} T^{11} + 233305356 p^{20} T^{12} + 11344 p^{24} T^{13} + p^{28} T^{14} )^{2} \) | |
| 89 | \( 1 - 18344 T + 168251168 T^{2} - 2231240410536 T^{3} + 34894610989575195 T^{4} - \)\(34\!\cdots\!96\)\( T^{5} + \)\(30\!\cdots\!12\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!89\)\( T^{8} - \)\(26\!\cdots\!96\)\( T^{9} + \)\(23\!\cdots\!60\)\( T^{10} - \)\(20\!\cdots\!76\)\( T^{11} + \)\(16\!\cdots\!91\)\( T^{12} - \)\(13\!\cdots\!64\)\( T^{13} + \)\(10\!\cdots\!12\)\( T^{14} - \)\(13\!\cdots\!64\)\( p^{4} T^{15} + \)\(16\!\cdots\!91\)\( p^{8} T^{16} - \)\(20\!\cdots\!76\)\( p^{12} T^{17} + \)\(23\!\cdots\!60\)\( p^{16} T^{18} - \)\(26\!\cdots\!96\)\( p^{20} T^{19} + \)\(33\!\cdots\!89\)\( p^{24} T^{20} - \)\(33\!\cdots\!20\)\( p^{28} T^{21} + \)\(30\!\cdots\!12\)\( p^{32} T^{22} - \)\(34\!\cdots\!96\)\( p^{36} T^{23} + 34894610989575195 p^{40} T^{24} - 2231240410536 p^{44} T^{25} + 168251168 p^{48} T^{26} - 18344 p^{52} T^{27} + p^{56} T^{28} \) | |
| 97 | \( 1 - 23246 T + 270188258 T^{2} - 3143397759126 T^{3} + 43662720097035979 T^{4} - \)\(53\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!36\)\( T^{6} - \)\(64\!\cdots\!16\)\( T^{7} + \)\(84\!\cdots\!05\)\( T^{8} - \)\(87\!\cdots\!62\)\( T^{9} + \)\(80\!\cdots\!34\)\( T^{10} - \)\(84\!\cdots\!18\)\( T^{11} + \)\(91\!\cdots\!71\)\( T^{12} - \)\(84\!\cdots\!84\)\( T^{13} + \)\(74\!\cdots\!36\)\( T^{14} - \)\(84\!\cdots\!84\)\( p^{4} T^{15} + \)\(91\!\cdots\!71\)\( p^{8} T^{16} - \)\(84\!\cdots\!18\)\( p^{12} T^{17} + \)\(80\!\cdots\!34\)\( p^{16} T^{18} - \)\(87\!\cdots\!62\)\( p^{20} T^{19} + \)\(84\!\cdots\!05\)\( p^{24} T^{20} - \)\(64\!\cdots\!16\)\( p^{28} T^{21} + \)\(54\!\cdots\!36\)\( p^{32} T^{22} - \)\(53\!\cdots\!80\)\( p^{36} T^{23} + 43662720097035979 p^{40} T^{24} - 3143397759126 p^{44} T^{25} + 270188258 p^{48} T^{26} - 23246 p^{52} T^{27} + p^{56} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.22539491976696010957428205553, −4.21200169400078484732334506183, −4.14410873504473352233437347480, −3.62583029797854442425644190205, −3.49851286255482081567946123878, −3.38812507253055875028018114537, −3.30502618125434695341072048334, −3.24661176368031213867758462575, −3.24461337960995060654687334066, −2.97821727296221870851908581571, −2.86993108563118287267484350307, −2.72581795509727457304634876415, −2.63105084285629562408102229535, −2.58019232609086302700436690434, −2.26134698992156073196147102264, −1.99075078694443529883612269892, −1.82359154153499580321544085081, −1.81238752933347171560100041983, −1.71171471440997122665944071333, −1.34373344324753864800366262603, −1.20924063068972670625322754101, −0.930616653185245521675872236173, −0.899151816340347699194058690776, −0.59540671638606188166765059378, −0.29355083651182001917566892722, 0.29355083651182001917566892722, 0.59540671638606188166765059378, 0.899151816340347699194058690776, 0.930616653185245521675872236173, 1.20924063068972670625322754101, 1.34373344324753864800366262603, 1.71171471440997122665944071333, 1.81238752933347171560100041983, 1.82359154153499580321544085081, 1.99075078694443529883612269892, 2.26134698992156073196147102264, 2.58019232609086302700436690434, 2.63105084285629562408102229535, 2.72581795509727457304634876415, 2.86993108563118287267484350307, 2.97821727296221870851908581571, 3.24461337960995060654687334066, 3.24661176368031213867758462575, 3.30502618125434695341072048334, 3.38812507253055875028018114537, 3.49851286255482081567946123878, 3.62583029797854442425644190205, 4.14410873504473352233437347480, 4.21200169400078484732334506183, 4.22539491976696010957428205553
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.