Dirichlet series
| L(s) = 1 | − 16·2-s − 243·4-s − 998·5-s + 1.36e3·7-s + 4.64e3·8-s + 1.59e4·10-s − 1.62e3·11-s + 1.35e4·13-s − 2.17e4·14-s + 3.27e4·16-s − 1.10e5·17-s + 1.05e5·19-s + 2.42e5·20-s + 2.59e4·22-s − 1.60e5·23-s + 1.25e5·25-s − 2.16e5·26-s − 3.30e5·28-s − 2.85e5·29-s − 9.96e4·31-s − 6.34e5·32-s + 1.77e6·34-s − 1.35e6·35-s + 1.76e5·37-s − 1.68e6·38-s − 4.63e6·40-s + 4.10e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.89·4-s − 3.57·5-s + 1.49·7-s + 3.20·8-s + 5.04·10-s − 0.366·11-s + 1.71·13-s − 2.11·14-s + 2.00·16-s − 5.47·17-s + 3.51·19-s + 6.77·20-s + 0.518·22-s − 2.74·23-s + 1.60·25-s − 2.41·26-s − 2.84·28-s − 2.17·29-s − 0.600·31-s − 3.42·32-s + 7.74·34-s − 5.35·35-s + 0.571·37-s − 4.96·38-s − 11.4·40-s + 0.929·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{26} \cdot 43^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{26} \cdot 43^{13}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(3^{26} \cdot 43^{13}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.17818\times 10^{27}\) |
| Root analytic conductor: | \(10.9951\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(13\) |
| Selberg data: | \((26,\ 3^{26} \cdot 43^{13} ,\ ( \ : [7/2]^{13} ),\ -1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 43 | \( ( 1 + p^{3} T )^{13} \) | |
| good | 2 | \( 1 + p^{4} T + 499 T^{2} + 3613 p T^{3} + 64877 p T^{4} + 405777 p^{2} T^{5} + 6265495 p^{2} T^{6} + 37718987 p^{3} T^{7} + 135023115 p^{5} T^{8} + 837154915 p^{6} T^{9} + 10730204725 p^{6} T^{10} + 64273522057 p^{7} T^{11} + 23846127347 p^{12} T^{12} + 268855436977 p^{12} T^{13} + 23846127347 p^{19} T^{14} + 64273522057 p^{21} T^{15} + 10730204725 p^{27} T^{16} + 837154915 p^{34} T^{17} + 135023115 p^{40} T^{18} + 37718987 p^{45} T^{19} + 6265495 p^{51} T^{20} + 405777 p^{58} T^{21} + 64877 p^{64} T^{22} + 3613 p^{71} T^{23} + 499 p^{77} T^{24} + p^{88} T^{25} + p^{91} T^{26} \) |
| 5 | \( 1 + 998 T + 174148 p T^{2} + 535689006 T^{3} + 300509982542 T^{4} + 28693425481694 p T^{5} + 2560442089648656 p^{2} T^{6} + 206419755308243666 p^{3} T^{7} + 15755669023851882384 p^{4} T^{8} + \)\(11\!\cdots\!22\)\( p^{5} T^{9} + \)\(75\!\cdots\!99\)\( p^{6} T^{10} + \)\(47\!\cdots\!32\)\( p^{7} T^{11} + \)\(28\!\cdots\!26\)\( p^{8} T^{12} + \)\(16\!\cdots\!08\)\( p^{9} T^{13} + \)\(28\!\cdots\!26\)\( p^{15} T^{14} + \)\(47\!\cdots\!32\)\( p^{21} T^{15} + \)\(75\!\cdots\!99\)\( p^{27} T^{16} + \)\(11\!\cdots\!22\)\( p^{33} T^{17} + 15755669023851882384 p^{39} T^{18} + 206419755308243666 p^{45} T^{19} + 2560442089648656 p^{51} T^{20} + 28693425481694 p^{57} T^{21} + 300509982542 p^{63} T^{22} + 535689006 p^{70} T^{23} + 174148 p^{78} T^{24} + 998 p^{84} T^{25} + p^{91} T^{26} \) | |
| 7 | \( 1 - 1360 T + 7058509 T^{2} - 7851071624 T^{3} + 22977827763868 T^{4} - 21067270496738864 T^{5} + 6663944414069580156 p T^{6} - \)\(35\!\cdots\!92\)\( T^{7} + \)\(68\!\cdots\!45\)\( T^{8} - \)\(44\!\cdots\!60\)\( T^{9} + \)\(78\!\cdots\!53\)\( T^{10} - \)\(44\!\cdots\!96\)\( T^{11} + \)\(74\!\cdots\!48\)\( T^{12} - \)\(56\!\cdots\!76\)\( p T^{13} + \)\(74\!\cdots\!48\)\( p^{7} T^{14} - \)\(44\!\cdots\!96\)\( p^{14} T^{15} + \)\(78\!\cdots\!53\)\( p^{21} T^{16} - \)\(44\!\cdots\!60\)\( p^{28} T^{17} + \)\(68\!\cdots\!45\)\( p^{35} T^{18} - \)\(35\!\cdots\!92\)\( p^{42} T^{19} + 6663944414069580156 p^{50} T^{20} - 21067270496738864 p^{56} T^{21} + 22977827763868 p^{63} T^{22} - 7851071624 p^{70} T^{23} + 7058509 p^{77} T^{24} - 1360 p^{84} T^{25} + p^{91} T^{26} \) | |
| 11 | \( 1 + 1620 T + 132588019 T^{2} + 123981604500 T^{3} + 7703760283965880 T^{4} + 3713200717818796164 T^{5} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!44\)\( T^{8} + \)\(83\!\cdots\!32\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} + \)\(37\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} + \)\(87\!\cdots\!20\)\( p T^{13} + \)\(11\!\cdots\!86\)\( p^{7} T^{14} + \)\(37\!\cdots\!84\)\( p^{14} T^{15} + \)\(85\!\cdots\!00\)\( p^{21} T^{16} + \)\(83\!\cdots\!32\)\( p^{28} T^{17} + \)\(55\!\cdots\!44\)\( p^{35} T^{18} + \)\(12\!\cdots\!20\)\( p^{42} T^{19} + \)\(25\!\cdots\!00\)\( p^{49} T^{20} + 3713200717818796164 p^{56} T^{21} + 7703760283965880 p^{63} T^{22} + 123981604500 p^{70} T^{23} + 132588019 p^{77} T^{24} + 1620 p^{84} T^{25} + p^{91} T^{26} \) | |
| 13 | \( 1 - 13550 T + 451004117 T^{2} - 4457271716692 T^{3} + 93112309019999992 T^{4} - \)\(74\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} - \)\(87\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!96\)\( T^{8} - \)\(81\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!86\)\( T^{12} - \)\(42\!\cdots\!16\)\( T^{13} + \)\(82\!\cdots\!86\)\( p^{7} T^{14} - \)\(63\!\cdots\!80\)\( p^{14} T^{15} + \)\(11\!\cdots\!04\)\( p^{21} T^{16} - \)\(81\!\cdots\!96\)\( p^{28} T^{17} + \)\(13\!\cdots\!96\)\( p^{35} T^{18} - \)\(87\!\cdots\!48\)\( p^{42} T^{19} + \)\(12\!\cdots\!04\)\( p^{49} T^{20} - \)\(74\!\cdots\!76\)\( p^{56} T^{21} + 93112309019999992 p^{63} T^{22} - 4457271716692 p^{70} T^{23} + 451004117 p^{77} T^{24} - 13550 p^{84} T^{25} + p^{91} T^{26} \) | |
| 17 | \( 1 + 110880 T + 7269241380 T^{2} + 350308303802712 T^{3} + 13915691873194028506 T^{4} + \)\(48\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(43\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!91\)\( T^{8} + \)\(30\!\cdots\!04\)\( T^{9} + \)\(72\!\cdots\!29\)\( T^{10} + \)\(16\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} + \)\(74\!\cdots\!34\)\( T^{13} + \)\(35\!\cdots\!06\)\( p^{7} T^{14} + \)\(16\!\cdots\!04\)\( p^{14} T^{15} + \)\(72\!\cdots\!29\)\( p^{21} T^{16} + \)\(30\!\cdots\!04\)\( p^{28} T^{17} + \)\(11\!\cdots\!91\)\( p^{35} T^{18} + \)\(43\!\cdots\!00\)\( p^{42} T^{19} + \)\(15\!\cdots\!04\)\( p^{49} T^{20} + \)\(48\!\cdots\!14\)\( p^{56} T^{21} + 13915691873194028506 p^{63} T^{22} + 350308303802712 p^{70} T^{23} + 7269241380 p^{77} T^{24} + 110880 p^{84} T^{25} + p^{91} T^{26} \) | |
| 19 | \( 1 - 105058 T + 10894929034 T^{2} - 729074101697894 T^{3} + 46770979546573980358 T^{4} - \)\(24\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!42\)\( T^{6} - \)\(27\!\cdots\!94\)\( p T^{7} + \)\(22\!\cdots\!20\)\( T^{8} - \)\(85\!\cdots\!82\)\( T^{9} + \)\(31\!\cdots\!91\)\( T^{10} - \)\(10\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!64\)\( p^{7} T^{14} - \)\(10\!\cdots\!04\)\( p^{14} T^{15} + \)\(31\!\cdots\!91\)\( p^{21} T^{16} - \)\(85\!\cdots\!82\)\( p^{28} T^{17} + \)\(22\!\cdots\!20\)\( p^{35} T^{18} - \)\(27\!\cdots\!94\)\( p^{43} T^{19} + \)\(12\!\cdots\!42\)\( p^{49} T^{20} - \)\(24\!\cdots\!36\)\( p^{56} T^{21} + 46770979546573980358 p^{63} T^{22} - 729074101697894 p^{70} T^{23} + 10894929034 p^{77} T^{24} - 105058 p^{84} T^{25} + p^{91} T^{26} \) | |
| 23 | \( 1 + 160184 T + 36979701828 T^{2} + 4753092322433576 T^{3} + \)\(65\!\cdots\!70\)\( T^{4} + \)\(68\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!92\)\( T^{6} + \)\(64\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!51\)\( T^{8} + \)\(42\!\cdots\!96\)\( T^{9} + \)\(13\!\cdots\!05\)\( p T^{10} + \)\(21\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} + \)\(83\!\cdots\!40\)\( T^{13} + \)\(13\!\cdots\!02\)\( p^{7} T^{14} + \)\(21\!\cdots\!56\)\( p^{14} T^{15} + \)\(13\!\cdots\!05\)\( p^{22} T^{16} + \)\(42\!\cdots\!96\)\( p^{28} T^{17} + \)\(55\!\cdots\!51\)\( p^{35} T^{18} + \)\(64\!\cdots\!20\)\( p^{42} T^{19} + \)\(71\!\cdots\!92\)\( p^{49} T^{20} + \)\(68\!\cdots\!64\)\( p^{56} T^{21} + \)\(65\!\cdots\!70\)\( p^{63} T^{22} + 4753092322433576 p^{70} T^{23} + 36979701828 p^{77} T^{24} + 160184 p^{84} T^{25} + p^{91} T^{26} \) | |
| 29 | \( 1 + 285546 T + 180604853400 T^{2} + 41013344500675266 T^{3} + \)\(14\!\cdots\!66\)\( T^{4} + \)\(27\!\cdots\!34\)\( T^{5} + \)\(71\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!24\)\( T^{8} + \)\(32\!\cdots\!66\)\( T^{9} + \)\(61\!\cdots\!47\)\( T^{10} + \)\(73\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!62\)\( T^{12} + \)\(13\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!62\)\( p^{7} T^{14} + \)\(73\!\cdots\!48\)\( p^{14} T^{15} + \)\(61\!\cdots\!47\)\( p^{21} T^{16} + \)\(32\!\cdots\!66\)\( p^{28} T^{17} + \)\(24\!\cdots\!24\)\( p^{35} T^{18} + \)\(11\!\cdots\!42\)\( p^{42} T^{19} + \)\(71\!\cdots\!40\)\( p^{49} T^{20} + \)\(27\!\cdots\!34\)\( p^{56} T^{21} + \)\(14\!\cdots\!66\)\( p^{63} T^{22} + 41013344500675266 p^{70} T^{23} + 180604853400 p^{77} T^{24} + 285546 p^{84} T^{25} + p^{91} T^{26} \) | |
| 31 | \( 1 + 99616 T + 241614413464 T^{2} + 18338344889599376 T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!92\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!72\)\( T^{9} + \)\(42\!\cdots\!31\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(14\!\cdots\!42\)\( p^{7} T^{14} + \)\(15\!\cdots\!08\)\( p^{14} T^{15} + \)\(42\!\cdots\!31\)\( p^{21} T^{16} + \)\(16\!\cdots\!72\)\( p^{28} T^{17} + \)\(10\!\cdots\!63\)\( p^{35} T^{18} + \)\(65\!\cdots\!12\)\( p^{42} T^{19} + \)\(20\!\cdots\!92\)\( p^{49} T^{20} + \)\(14\!\cdots\!12\)\( p^{56} T^{21} + \)\(27\!\cdots\!10\)\( p^{63} T^{22} + 18338344889599376 p^{70} T^{23} + 241614413464 p^{77} T^{24} + 99616 p^{84} T^{25} + p^{91} T^{26} \) | |
| 37 | \( 1 - 176038 T + 1002592896582 T^{2} - 172621415570488610 T^{3} + \)\(48\!\cdots\!62\)\( T^{4} - \)\(79\!\cdots\!90\)\( T^{5} + \)\(14\!\cdots\!62\)\( T^{6} - \)\(22\!\cdots\!06\)\( T^{7} + \)\(31\!\cdots\!64\)\( T^{8} - \)\(45\!\cdots\!18\)\( T^{9} + \)\(50\!\cdots\!45\)\( T^{10} - \)\(65\!\cdots\!96\)\( T^{11} + \)\(61\!\cdots\!44\)\( T^{12} - \)\(71\!\cdots\!12\)\( T^{13} + \)\(61\!\cdots\!44\)\( p^{7} T^{14} - \)\(65\!\cdots\!96\)\( p^{14} T^{15} + \)\(50\!\cdots\!45\)\( p^{21} T^{16} - \)\(45\!\cdots\!18\)\( p^{28} T^{17} + \)\(31\!\cdots\!64\)\( p^{35} T^{18} - \)\(22\!\cdots\!06\)\( p^{42} T^{19} + \)\(14\!\cdots\!62\)\( p^{49} T^{20} - \)\(79\!\cdots\!90\)\( p^{56} T^{21} + \)\(48\!\cdots\!62\)\( p^{63} T^{22} - 172621415570488610 p^{70} T^{23} + 1002592896582 p^{77} T^{24} - 176038 p^{84} T^{25} + p^{91} T^{26} \) | |
| 41 | \( 1 - 410260 T + 31116447684 p T^{2} - 520566650539856152 T^{3} + \)\(84\!\cdots\!34\)\( T^{4} - \)\(33\!\cdots\!10\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!59\)\( T^{8} - \)\(44\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!41\)\( T^{10} - \)\(11\!\cdots\!44\)\( T^{11} + \)\(73\!\cdots\!58\)\( T^{12} - \)\(23\!\cdots\!38\)\( T^{13} + \)\(73\!\cdots\!58\)\( p^{7} T^{14} - \)\(11\!\cdots\!44\)\( p^{14} T^{15} + \)\(32\!\cdots\!41\)\( p^{21} T^{16} - \)\(44\!\cdots\!08\)\( p^{28} T^{17} + \)\(12\!\cdots\!59\)\( p^{35} T^{18} - \)\(14\!\cdots\!96\)\( p^{42} T^{19} + \)\(37\!\cdots\!04\)\( p^{49} T^{20} - \)\(33\!\cdots\!10\)\( p^{56} T^{21} + \)\(84\!\cdots\!34\)\( p^{63} T^{22} - 520566650539856152 p^{70} T^{23} + 31116447684 p^{78} T^{24} - 410260 p^{84} T^{25} + p^{91} T^{26} \) | |
| 47 | \( 1 - 424556 T + 3688252683440 T^{2} - 2420585272729575804 T^{3} + \)\(67\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!12\)\( T^{6} - \)\(77\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!18\)\( T^{8} - \)\(74\!\cdots\!56\)\( T^{9} + \)\(64\!\cdots\!19\)\( T^{10} - \)\(53\!\cdots\!64\)\( T^{11} + \)\(40\!\cdots\!52\)\( T^{12} - \)\(30\!\cdots\!32\)\( T^{13} + \)\(40\!\cdots\!52\)\( p^{7} T^{14} - \)\(53\!\cdots\!64\)\( p^{14} T^{15} + \)\(64\!\cdots\!19\)\( p^{21} T^{16} - \)\(74\!\cdots\!56\)\( p^{28} T^{17} + \)\(81\!\cdots\!18\)\( p^{35} T^{18} - \)\(77\!\cdots\!80\)\( p^{42} T^{19} + \)\(83\!\cdots\!12\)\( p^{49} T^{20} - \)\(56\!\cdots\!60\)\( p^{56} T^{21} + \)\(67\!\cdots\!76\)\( p^{63} T^{22} - 2420585272729575804 p^{70} T^{23} + 3688252683440 p^{77} T^{24} - 424556 p^{84} T^{25} + p^{91} T^{26} \) | |
| 53 | \( 1 + 3992458 T + 15589250135817 T^{2} + 39082970394967823704 T^{3} + \)\(92\!\cdots\!16\)\( T^{4} + \)\(17\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} + \)\(49\!\cdots\!88\)\( T^{7} + \)\(73\!\cdots\!96\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(16\!\cdots\!28\)\( T^{11} + \)\(19\!\cdots\!82\)\( T^{12} + \)\(21\!\cdots\!60\)\( T^{13} + \)\(19\!\cdots\!82\)\( p^{7} T^{14} + \)\(16\!\cdots\!28\)\( p^{14} T^{15} + \)\(13\!\cdots\!44\)\( p^{21} T^{16} + \)\(10\!\cdots\!68\)\( p^{28} T^{17} + \)\(73\!\cdots\!96\)\( p^{35} T^{18} + \)\(49\!\cdots\!88\)\( p^{42} T^{19} + \)\(31\!\cdots\!44\)\( p^{49} T^{20} + \)\(17\!\cdots\!84\)\( p^{56} T^{21} + \)\(92\!\cdots\!16\)\( p^{63} T^{22} + 39082970394967823704 p^{70} T^{23} + 15589250135817 p^{77} T^{24} + 3992458 p^{84} T^{25} + p^{91} T^{26} \) | |
| 59 | \( 1 + 2248836 T + 18986189218171 T^{2} + 636896445896421216 p T^{3} + \)\(17\!\cdots\!18\)\( T^{4} + \)\(31\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!54\)\( T^{6} + \)\(17\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!59\)\( T^{8} + \)\(76\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!17\)\( T^{10} + \)\(25\!\cdots\!88\)\( T^{11} + \)\(56\!\cdots\!40\)\( T^{12} + \)\(70\!\cdots\!64\)\( T^{13} + \)\(56\!\cdots\!40\)\( p^{7} T^{14} + \)\(25\!\cdots\!88\)\( p^{14} T^{15} + \)\(18\!\cdots\!17\)\( p^{21} T^{16} + \)\(76\!\cdots\!40\)\( p^{28} T^{17} + \)\(51\!\cdots\!59\)\( p^{35} T^{18} + \)\(17\!\cdots\!32\)\( p^{42} T^{19} + \)\(11\!\cdots\!54\)\( p^{49} T^{20} + \)\(31\!\cdots\!68\)\( p^{56} T^{21} + \)\(17\!\cdots\!18\)\( p^{63} T^{22} + 636896445896421216 p^{71} T^{23} + 18986189218171 p^{77} T^{24} + 2248836 p^{84} T^{25} + p^{91} T^{26} \) | |
| 61 | \( 1 - 6210394 T + 38426666503003 T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!92\)\( T^{5} + \)\(56\!\cdots\!64\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!17\)\( T^{8} - \)\(83\!\cdots\!78\)\( T^{9} + \)\(18\!\cdots\!55\)\( T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{13} + \)\(70\!\cdots\!72\)\( p^{7} T^{14} - \)\(36\!\cdots\!04\)\( p^{14} T^{15} + \)\(18\!\cdots\!55\)\( p^{21} T^{16} - \)\(83\!\cdots\!78\)\( p^{28} T^{17} + \)\(36\!\cdots\!17\)\( p^{35} T^{18} - \)\(14\!\cdots\!80\)\( p^{42} T^{19} + \)\(56\!\cdots\!64\)\( p^{49} T^{20} - \)\(18\!\cdots\!92\)\( p^{56} T^{21} + \)\(60\!\cdots\!88\)\( p^{63} T^{22} - \)\(15\!\cdots\!76\)\( p^{70} T^{23} + 38426666503003 p^{77} T^{24} - 6210394 p^{84} T^{25} + p^{91} T^{26} \) | |
| 67 | \( 1 + 1993648 T + 36332499483043 T^{2} + 80369020063411411764 T^{3} + \)\(68\!\cdots\!56\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(90\!\cdots\!36\)\( T^{8} + \)\(22\!\cdots\!72\)\( T^{9} + \)\(75\!\cdots\!16\)\( T^{10} + \)\(18\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!18\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{13} + \)\(53\!\cdots\!18\)\( p^{7} T^{14} + \)\(18\!\cdots\!92\)\( p^{14} T^{15} + \)\(75\!\cdots\!16\)\( p^{21} T^{16} + \)\(22\!\cdots\!72\)\( p^{28} T^{17} + \)\(90\!\cdots\!36\)\( p^{35} T^{18} + \)\(22\!\cdots\!60\)\( p^{42} T^{19} + \)\(89\!\cdots\!48\)\( p^{49} T^{20} + \)\(16\!\cdots\!32\)\( p^{56} T^{21} + \)\(68\!\cdots\!56\)\( p^{63} T^{22} + 80369020063411411764 p^{70} T^{23} + 36332499483043 p^{77} T^{24} + 1993648 p^{84} T^{25} + p^{91} T^{26} \) | |
| 71 | \( 1 + 4978064 T + 87202612193659 T^{2} + 5528935375628066240 p T^{3} + \)\(37\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!70\)\( T^{6} + \)\(37\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!19\)\( T^{8} + \)\(66\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!13\)\( T^{10} + \)\(88\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!08\)\( T^{12} + \)\(91\!\cdots\!08\)\( T^{13} + \)\(34\!\cdots\!08\)\( p^{7} T^{14} + \)\(88\!\cdots\!48\)\( p^{14} T^{15} + \)\(29\!\cdots\!13\)\( p^{21} T^{16} + \)\(66\!\cdots\!92\)\( p^{28} T^{17} + \)\(20\!\cdots\!19\)\( p^{35} T^{18} + \)\(37\!\cdots\!88\)\( p^{42} T^{19} + \)\(10\!\cdots\!70\)\( p^{49} T^{20} + \)\(15\!\cdots\!52\)\( p^{56} T^{21} + \)\(37\!\cdots\!18\)\( p^{63} T^{22} + 5528935375628066240 p^{71} T^{23} + 87202612193659 p^{77} T^{24} + 4978064 p^{84} T^{25} + p^{91} T^{26} \) | |
| 73 | \( 1 - 8224814 T + 92481350221839 T^{2} - \)\(53\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!04\)\( T^{4} - \)\(18\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} - \)\(85\!\cdots\!50\)\( T^{9} + \)\(36\!\cdots\!23\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} + \)\(49\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + \)\(49\!\cdots\!68\)\( p^{7} T^{14} - \)\(12\!\cdots\!68\)\( p^{14} T^{15} + \)\(36\!\cdots\!23\)\( p^{21} T^{16} - \)\(85\!\cdots\!50\)\( p^{28} T^{17} + \)\(22\!\cdots\!37\)\( p^{35} T^{18} - \)\(45\!\cdots\!64\)\( p^{42} T^{19} + \)\(10\!\cdots\!88\)\( p^{49} T^{20} - \)\(18\!\cdots\!96\)\( p^{56} T^{21} + \)\(38\!\cdots\!04\)\( p^{63} T^{22} - \)\(53\!\cdots\!16\)\( p^{70} T^{23} + 92481350221839 p^{77} T^{24} - 8224814 p^{84} T^{25} + p^{91} T^{26} \) | |
| 79 | \( 1 - 6945708 T + 117121136837052 T^{2} - \)\(56\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(72\!\cdots\!74\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + \)\(18\!\cdots\!35\)\( T^{10} - \)\(46\!\cdots\!04\)\( T^{11} + \)\(42\!\cdots\!92\)\( T^{12} - \)\(97\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!92\)\( p^{7} T^{14} - \)\(46\!\cdots\!04\)\( p^{14} T^{15} + \)\(18\!\cdots\!35\)\( p^{21} T^{16} - \)\(19\!\cdots\!48\)\( p^{28} T^{17} + \)\(72\!\cdots\!74\)\( p^{35} T^{18} - \)\(74\!\cdots\!48\)\( p^{42} T^{19} + \)\(24\!\cdots\!68\)\( p^{49} T^{20} - \)\(24\!\cdots\!60\)\( p^{56} T^{21} + \)\(65\!\cdots\!68\)\( p^{63} T^{22} - \)\(56\!\cdots\!28\)\( p^{70} T^{23} + 117121136837052 p^{77} T^{24} - 6945708 p^{84} T^{25} + p^{91} T^{26} \) | |
| 83 | \( 1 + 22937328 T + 403848212345231 T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} + \)\(35\!\cdots\!72\)\( T^{5} + \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + \)\(83\!\cdots\!60\)\( T^{8} + \)\(44\!\cdots\!24\)\( T^{9} + \)\(27\!\cdots\!48\)\( T^{10} + \)\(15\!\cdots\!32\)\( T^{11} + \)\(97\!\cdots\!38\)\( T^{12} + \)\(51\!\cdots\!32\)\( T^{13} + \)\(97\!\cdots\!38\)\( p^{7} T^{14} + \)\(15\!\cdots\!32\)\( p^{14} T^{15} + \)\(27\!\cdots\!48\)\( p^{21} T^{16} + \)\(44\!\cdots\!24\)\( p^{28} T^{17} + \)\(83\!\cdots\!60\)\( p^{35} T^{18} + \)\(14\!\cdots\!68\)\( p^{42} T^{19} + \)\(24\!\cdots\!76\)\( p^{49} T^{20} + \)\(35\!\cdots\!72\)\( p^{56} T^{21} + \)\(46\!\cdots\!36\)\( p^{63} T^{22} + \)\(46\!\cdots\!40\)\( p^{70} T^{23} + 403848212345231 p^{77} T^{24} + 22937328 p^{84} T^{25} + p^{91} T^{26} \) | |
| 89 | \( 1 + 9291302 T + 323406546094099 T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(56\!\cdots\!92\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!68\)\( T^{6} + \)\(56\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!61\)\( T^{8} + \)\(45\!\cdots\!46\)\( T^{9} + \)\(42\!\cdots\!63\)\( T^{10} + \)\(28\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!60\)\( T^{12} + \)\(14\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!60\)\( p^{7} T^{14} + \)\(28\!\cdots\!48\)\( p^{14} T^{15} + \)\(42\!\cdots\!63\)\( p^{21} T^{16} + \)\(45\!\cdots\!46\)\( p^{28} T^{17} + \)\(61\!\cdots\!61\)\( p^{35} T^{18} + \)\(56\!\cdots\!24\)\( p^{42} T^{19} + \)\(68\!\cdots\!68\)\( p^{49} T^{20} + \)\(51\!\cdots\!56\)\( p^{56} T^{21} + \)\(56\!\cdots\!92\)\( p^{63} T^{22} + \)\(31\!\cdots\!04\)\( p^{70} T^{23} + 323406546094099 p^{77} T^{24} + 9291302 p^{84} T^{25} + p^{91} T^{26} \) | |
| 97 | \( 1 - 10001852 T + 551581550822996 T^{2} - \)\(38\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(71\!\cdots\!70\)\( p T^{5} + \)\(23\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!08\)\( T^{7} + \)\(31\!\cdots\!27\)\( T^{8} - \)\(74\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!81\)\( T^{10} - \)\(66\!\cdots\!32\)\( T^{11} + \)\(32\!\cdots\!38\)\( T^{12} - \)\(56\!\cdots\!82\)\( T^{13} + \)\(32\!\cdots\!38\)\( p^{7} T^{14} - \)\(66\!\cdots\!32\)\( p^{14} T^{15} + \)\(34\!\cdots\!81\)\( p^{21} T^{16} - \)\(74\!\cdots\!80\)\( p^{28} T^{17} + \)\(31\!\cdots\!27\)\( p^{35} T^{18} - \)\(80\!\cdots\!08\)\( p^{42} T^{19} + \)\(23\!\cdots\!80\)\( p^{49} T^{20} - \)\(71\!\cdots\!70\)\( p^{57} T^{21} + \)\(14\!\cdots\!86\)\( p^{63} T^{22} - \)\(38\!\cdots\!08\)\( p^{70} T^{23} + 551581550822996 p^{77} T^{24} - 10001852 p^{84} T^{25} + p^{91} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.05461762310461832453955410451, −2.90927867482978221681082785706, −2.88353758230972069285664864676, −2.73891479822798397791358989687, −2.70377658364808323281908884965, −2.40118614154969101758647349965, −2.35513531979109964346461875757, −2.27487809424816993348349763588, −2.25802600719392982546665802400, −2.09023473161212212623582314343, −2.03033029217755371205754674459, −1.94281662587282911271902590381, −1.85597619390279506440798998679, −1.71896771378764195089523969720, −1.58483051618166992724270485627, −1.51824934406878513884066386994, −1.46456176219286545739061419179, −1.25646404574567169880882976236, −1.18658828312449388506593558455, −1.17288161268328394555395624361, −1.06811791037677096247122582780, −0.924867206244969590485219580196, −0.894222761137565858921941206317, −0.873656726435622578535855064395, −0.76805713188906949929603809059, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.76805713188906949929603809059, 0.873656726435622578535855064395, 0.894222761137565858921941206317, 0.924867206244969590485219580196, 1.06811791037677096247122582780, 1.17288161268328394555395624361, 1.18658828312449388506593558455, 1.25646404574567169880882976236, 1.46456176219286545739061419179, 1.51824934406878513884066386994, 1.58483051618166992724270485627, 1.71896771378764195089523969720, 1.85597619390279506440798998679, 1.94281662587282911271902590381, 2.03033029217755371205754674459, 2.09023473161212212623582314343, 2.25802600719392982546665802400, 2.27487809424816993348349763588, 2.35513531979109964346461875757, 2.40118614154969101758647349965, 2.70377658364808323281908884965, 2.73891479822798397791358989687, 2.88353758230972069285664864676, 2.90927867482978221681082785706, 3.05461762310461832453955410451
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.