Dirichlet series
| L(s) = 1 | − 2·2-s − 54·3-s + 23·4-s − 100·5-s + 108·6-s − 162·8-s + 1.21e3·9-s + 200·10-s − 604·11-s − 1.24e3·12-s + 2.70e3·13-s + 5.40e3·15-s + 556·16-s − 3.02e3·17-s − 2.43e3·18-s − 1.72e3·19-s − 2.30e3·20-s + 1.20e3·22-s + 4.48e3·23-s + 8.74e3·24-s + 1.19e4·25-s − 5.40e3·26-s − 1.02e4·27-s − 1.06e4·29-s − 1.08e4·30-s − 3.97e3·31-s + 3.42e3·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 3.46·3-s + 0.718·4-s − 1.78·5-s + 1.22·6-s − 0.894·8-s + 5·9-s + 0.632·10-s − 1.50·11-s − 2.48·12-s + 4.43·13-s + 6.19·15-s + 0.542·16-s − 2.54·17-s − 1.76·18-s − 1.09·19-s − 1.28·20-s + 0.532·22-s + 1.76·23-s + 3.10·24-s + 3.83·25-s − 1.56·26-s − 2.69·27-s − 2.34·29-s − 2.19·30-s − 0.743·31-s + 0.591·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.94940\times 10^{16}\) |
| Root analytic conductor: | \(4.85555\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 7^{24} ,\ ( \ : [5/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(1.010447594\) |
| \(L(\frac12)\) | \(\approx\) | \(1.010447594\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{2} T + p^{4} T^{2} )^{6} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T - 19 T^{2} + 39 p T^{3} + 361 T^{4} - 543 p^{4} T^{5} - 3627 p^{3} T^{6} + 1949 p^{5} T^{7} - 49437 p^{4} T^{8} - 24281 p^{8} T^{9} + 336121 p^{7} T^{10} + 392339 p^{9} T^{11} + 74353 p^{8} T^{12} + 392339 p^{14} T^{13} + 336121 p^{17} T^{14} - 24281 p^{23} T^{15} - 49437 p^{24} T^{16} + 1949 p^{30} T^{17} - 3627 p^{33} T^{18} - 543 p^{39} T^{19} + 361 p^{40} T^{20} + 39 p^{46} T^{21} - 19 p^{50} T^{22} + p^{56} T^{23} + p^{60} T^{24} \) |
| 5 | \( 1 + 4 p^{2} T - 1972 T^{2} - 625176 T^{3} - 19218683 T^{4} + 9934368 p^{3} T^{5} + 80453228244 T^{6} - 542591820044 p T^{7} - 312665399771082 T^{8} + 17122648366076 T^{9} + 703325585973913292 T^{10} + 17913380491466957168 T^{11} - \)\(16\!\cdots\!31\)\( T^{12} + 17913380491466957168 p^{5} T^{13} + 703325585973913292 p^{10} T^{14} + 17122648366076 p^{15} T^{15} - 312665399771082 p^{20} T^{16} - 542591820044 p^{26} T^{17} + 80453228244 p^{30} T^{18} + 9934368 p^{38} T^{19} - 19218683 p^{40} T^{20} - 625176 p^{45} T^{21} - 1972 p^{50} T^{22} + 4 p^{57} T^{23} + p^{60} T^{24} \) | |
| 11 | \( 1 + 604 T - 100654 T^{2} + 28298752 T^{3} + 85749678749 T^{4} - 9180899434392 T^{5} - 4676434772779994 T^{6} + 9575566634437123572 T^{7} + \)\(61\!\cdots\!82\)\( T^{8} - \)\(91\!\cdots\!96\)\( T^{9} + \)\(63\!\cdots\!66\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{11} - \)\(91\!\cdots\!55\)\( T^{12} + \)\(14\!\cdots\!12\)\( p^{5} T^{13} + \)\(63\!\cdots\!66\)\( p^{10} T^{14} - \)\(91\!\cdots\!96\)\( p^{15} T^{15} + \)\(61\!\cdots\!82\)\( p^{20} T^{16} + 9575566634437123572 p^{25} T^{17} - 4676434772779994 p^{30} T^{18} - 9180899434392 p^{35} T^{19} + 85749678749 p^{40} T^{20} + 28298752 p^{45} T^{21} - 100654 p^{50} T^{22} + 604 p^{55} T^{23} + p^{60} T^{24} \) | |
| 13 | \( ( 1 - 8 p^{2} T + 2101160 T^{2} - 1701654952 T^{3} + 1567459550619 T^{4} - 957961480510864 T^{5} + 695838444947896272 T^{6} - 957961480510864 p^{5} T^{7} + 1567459550619 p^{10} T^{8} - 1701654952 p^{15} T^{9} + 2101160 p^{20} T^{10} - 8 p^{27} T^{11} + p^{30} T^{12} )^{2} \) | |
| 17 | \( 1 + 3028 T - 311260 T^{2} - 4442310776 T^{3} + 9583852492493 T^{4} + 13084924900221120 T^{5} - 20705734738250922596 T^{6} - \)\(11\!\cdots\!68\)\( T^{7} + \)\(38\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!40\)\( T^{9} - \)\(60\!\cdots\!04\)\( T^{10} - \)\(38\!\cdots\!92\)\( p T^{11} + \)\(22\!\cdots\!93\)\( p^{2} T^{12} - \)\(38\!\cdots\!92\)\( p^{6} T^{13} - \)\(60\!\cdots\!04\)\( p^{10} T^{14} + \)\(18\!\cdots\!40\)\( p^{15} T^{15} + \)\(38\!\cdots\!58\)\( p^{20} T^{16} - \)\(11\!\cdots\!68\)\( p^{25} T^{17} - 20705734738250922596 p^{30} T^{18} + 13084924900221120 p^{35} T^{19} + 9583852492493 p^{40} T^{20} - 4442310776 p^{45} T^{21} - 311260 p^{50} T^{22} + 3028 p^{55} T^{23} + p^{60} T^{24} \) | |
| 19 | \( 1 + 1728 T - 3902010 T^{2} - 8190782208 T^{3} + 6228811311 p T^{4} + 219992056216704 p T^{5} + 9056902907985374274 T^{6} + \)\(22\!\cdots\!92\)\( T^{7} + \)\(33\!\cdots\!22\)\( T^{8} + \)\(49\!\cdots\!00\)\( T^{9} - \)\(17\!\cdots\!10\)\( T^{10} - \)\(66\!\cdots\!16\)\( T^{11} + \)\(48\!\cdots\!29\)\( T^{12} - \)\(66\!\cdots\!16\)\( p^{5} T^{13} - \)\(17\!\cdots\!10\)\( p^{10} T^{14} + \)\(49\!\cdots\!00\)\( p^{15} T^{15} + \)\(33\!\cdots\!22\)\( p^{20} T^{16} + \)\(22\!\cdots\!92\)\( p^{25} T^{17} + 9056902907985374274 p^{30} T^{18} + 219992056216704 p^{36} T^{19} + 6228811311 p^{41} T^{20} - 8190782208 p^{45} T^{21} - 3902010 p^{50} T^{22} + 1728 p^{55} T^{23} + p^{60} T^{24} \) | |
| 23 | \( 1 - 4484 T + 1240010 T^{2} + 23404017216 T^{3} - 48551175082187 T^{4} + 72136747394356008 T^{5} + \)\(31\!\cdots\!50\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} + \)\(83\!\cdots\!56\)\( T^{9} - \)\(71\!\cdots\!74\)\( T^{10} - \)\(14\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!41\)\( T^{12} - \)\(14\!\cdots\!52\)\( p^{5} T^{13} - \)\(71\!\cdots\!74\)\( p^{10} T^{14} + \)\(83\!\cdots\!56\)\( p^{15} T^{15} + \)\(11\!\cdots\!50\)\( p^{20} T^{16} - \)\(20\!\cdots\!00\)\( p^{25} T^{17} + \)\(31\!\cdots\!50\)\( p^{30} T^{18} + 72136747394356008 p^{35} T^{19} - 48551175082187 p^{40} T^{20} + 23404017216 p^{45} T^{21} + 1240010 p^{50} T^{22} - 4484 p^{55} T^{23} + p^{60} T^{24} \) | |
| 29 | \( ( 1 + 5320 T + 81920486 T^{2} + 415516942408 T^{3} + 3329157696001607 T^{4} + 14790454945150175696 T^{5} + \)\(84\!\cdots\!28\)\( T^{6} + 14790454945150175696 p^{5} T^{7} + 3329157696001607 p^{10} T^{8} + 415516942408 p^{15} T^{9} + 81920486 p^{20} T^{10} + 5320 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 31 | \( 1 + 3976 T - 67478290 T^{2} - 525972398624 T^{3} + 1443959460122069 T^{4} + 25314872084782944528 T^{5} + \)\(37\!\cdots\!86\)\( T^{6} - \)\(60\!\cdots\!12\)\( T^{7} - \)\(30\!\cdots\!34\)\( T^{8} + \)\(77\!\cdots\!88\)\( T^{9} + \)\(98\!\cdots\!82\)\( T^{10} - \)\(10\!\cdots\!04\)\( p T^{11} - \)\(28\!\cdots\!03\)\( p^{2} T^{12} - \)\(10\!\cdots\!04\)\( p^{6} T^{13} + \)\(98\!\cdots\!82\)\( p^{10} T^{14} + \)\(77\!\cdots\!88\)\( p^{15} T^{15} - \)\(30\!\cdots\!34\)\( p^{20} T^{16} - \)\(60\!\cdots\!12\)\( p^{25} T^{17} + \)\(37\!\cdots\!86\)\( p^{30} T^{18} + 25314872084782944528 p^{35} T^{19} + 1443959460122069 p^{40} T^{20} - 525972398624 p^{45} T^{21} - 67478290 p^{50} T^{22} + 3976 p^{55} T^{23} + p^{60} T^{24} \) | |
| 37 | \( 1 + 22680 T + 11052882 T^{2} - 1104088413536 T^{3} + 32776423396104429 T^{4} + \)\(27\!\cdots\!12\)\( T^{5} - \)\(27\!\cdots\!94\)\( T^{6} - \)\(60\!\cdots\!92\)\( T^{7} + \)\(29\!\cdots\!98\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{9} - \)\(22\!\cdots\!62\)\( T^{10} - \)\(19\!\cdots\!20\)\( T^{11} + \)\(92\!\cdots\!61\)\( T^{12} - \)\(19\!\cdots\!20\)\( p^{5} T^{13} - \)\(22\!\cdots\!62\)\( p^{10} T^{14} - \)\(25\!\cdots\!96\)\( p^{15} T^{15} + \)\(29\!\cdots\!98\)\( p^{20} T^{16} - \)\(60\!\cdots\!92\)\( p^{25} T^{17} - \)\(27\!\cdots\!94\)\( p^{30} T^{18} + \)\(27\!\cdots\!12\)\( p^{35} T^{19} + 32776423396104429 p^{40} T^{20} - 1104088413536 p^{45} T^{21} + 11052882 p^{50} T^{22} + 22680 p^{55} T^{23} + p^{60} T^{24} \) | |
| 41 | \( ( 1 - 28756 T + 750193916 T^{2} - 13026738107732 T^{3} + 211306160121247683 T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!92\)\( T^{6} - \)\(26\!\cdots\!00\)\( p^{5} T^{7} + 211306160121247683 p^{10} T^{8} - 13026738107732 p^{15} T^{9} + 750193916 p^{20} T^{10} - 28756 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 6768 T + 494671554 T^{2} + 2742913738512 T^{3} + 119159445546447735 T^{4} + \)\(53\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!64\)\( T^{6} + \)\(53\!\cdots\!92\)\( p^{5} T^{7} + 119159445546447735 p^{10} T^{8} + 2742913738512 p^{15} T^{9} + 494671554 p^{20} T^{10} + 6768 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 47 | \( 1 + 51552 T + 545768190 T^{2} - 13624313747328 T^{3} - 149921762605886475 T^{4} + \)\(81\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} - \)\(10\!\cdots\!32\)\( T^{7} - \)\(29\!\cdots\!38\)\( T^{8} + \)\(43\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!18\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{11} - \)\(21\!\cdots\!95\)\( T^{12} + \)\(12\!\cdots\!56\)\( p^{5} T^{13} + \)\(13\!\cdots\!18\)\( p^{10} T^{14} + \)\(43\!\cdots\!68\)\( p^{15} T^{15} - \)\(29\!\cdots\!38\)\( p^{20} T^{16} - \)\(10\!\cdots\!32\)\( p^{25} T^{17} + \)\(14\!\cdots\!06\)\( p^{30} T^{18} + \)\(81\!\cdots\!68\)\( p^{35} T^{19} - 149921762605886475 p^{40} T^{20} - 13624313747328 p^{45} T^{21} + 545768190 p^{50} T^{22} + 51552 p^{55} T^{23} + p^{60} T^{24} \) | |
| 53 | \( 1 + 80884 T + 2534342390 T^{2} + 41567177443168 T^{3} + 446298672567007037 T^{4} - \)\(54\!\cdots\!84\)\( T^{5} - \)\(68\!\cdots\!22\)\( T^{6} - \)\(22\!\cdots\!36\)\( T^{7} - \)\(36\!\cdots\!06\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{9} + \)\(37\!\cdots\!54\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{11} + \)\(94\!\cdots\!17\)\( T^{12} + \)\(32\!\cdots\!00\)\( p^{5} T^{13} + \)\(37\!\cdots\!54\)\( p^{10} T^{14} - \)\(30\!\cdots\!96\)\( p^{15} T^{15} - \)\(36\!\cdots\!06\)\( p^{20} T^{16} - \)\(22\!\cdots\!36\)\( p^{25} T^{17} - \)\(68\!\cdots\!22\)\( p^{30} T^{18} - \)\(54\!\cdots\!84\)\( p^{35} T^{19} + 446298672567007037 p^{40} T^{20} + 41567177443168 p^{45} T^{21} + 2534342390 p^{50} T^{22} + 80884 p^{55} T^{23} + p^{60} T^{24} \) | |
| 59 | \( 1 + 8872 T - 3103933402 T^{2} - 29903967828000 T^{3} + 5191481299129909741 T^{4} + \)\(49\!\cdots\!48\)\( T^{5} - \)\(60\!\cdots\!54\)\( T^{6} - \)\(53\!\cdots\!68\)\( T^{7} + \)\(55\!\cdots\!78\)\( T^{8} + \)\(62\!\cdots\!20\)\( p T^{9} - \)\(44\!\cdots\!42\)\( T^{10} - \)\(11\!\cdots\!68\)\( T^{11} + \)\(32\!\cdots\!09\)\( T^{12} - \)\(11\!\cdots\!68\)\( p^{5} T^{13} - \)\(44\!\cdots\!42\)\( p^{10} T^{14} + \)\(62\!\cdots\!20\)\( p^{16} T^{15} + \)\(55\!\cdots\!78\)\( p^{20} T^{16} - \)\(53\!\cdots\!68\)\( p^{25} T^{17} - \)\(60\!\cdots\!54\)\( p^{30} T^{18} + \)\(49\!\cdots\!48\)\( p^{35} T^{19} + 5191481299129909741 p^{40} T^{20} - 29903967828000 p^{45} T^{21} - 3103933402 p^{50} T^{22} + 8872 p^{55} T^{23} + p^{60} T^{24} \) | |
| 61 | \( 1 + 50896 T - 2255052424 T^{2} - 124663902915232 T^{3} + 4037518675087990533 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} - \)\(65\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!08\)\( T^{7} + \)\(95\!\cdots\!98\)\( T^{8} + \)\(12\!\cdots\!76\)\( T^{9} - \)\(10\!\cdots\!28\)\( T^{10} - \)\(39\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!97\)\( T^{12} - \)\(39\!\cdots\!00\)\( p^{5} T^{13} - \)\(10\!\cdots\!28\)\( p^{10} T^{14} + \)\(12\!\cdots\!76\)\( p^{15} T^{15} + \)\(95\!\cdots\!98\)\( p^{20} T^{16} - \)\(16\!\cdots\!08\)\( p^{25} T^{17} - \)\(65\!\cdots\!56\)\( p^{30} T^{18} + \)\(17\!\cdots\!96\)\( p^{35} T^{19} + 4037518675087990533 p^{40} T^{20} - 124663902915232 p^{45} T^{21} - 2255052424 p^{50} T^{22} + 50896 p^{55} T^{23} + p^{60} T^{24} \) | |
| 67 | \( 1 + 6480 T - 5057790354 T^{2} - 22860088380864 T^{3} + 12785183483613752157 T^{4} + \)\(32\!\cdots\!16\)\( T^{5} - \)\(23\!\cdots\!26\)\( T^{6} - \)\(19\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!22\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{9} - \)\(51\!\cdots\!06\)\( T^{10} + \)\(38\!\cdots\!92\)\( T^{11} + \)\(68\!\cdots\!33\)\( T^{12} + \)\(38\!\cdots\!92\)\( p^{5} T^{13} - \)\(51\!\cdots\!06\)\( p^{10} T^{14} + \)\(12\!\cdots\!12\)\( p^{15} T^{15} + \)\(37\!\cdots\!22\)\( p^{20} T^{16} - \)\(19\!\cdots\!40\)\( p^{25} T^{17} - \)\(23\!\cdots\!26\)\( p^{30} T^{18} + \)\(32\!\cdots\!16\)\( p^{35} T^{19} + 12785183483613752157 p^{40} T^{20} - 22860088380864 p^{45} T^{21} - 5057790354 p^{50} T^{22} + 6480 p^{55} T^{23} + p^{60} T^{24} \) | |
| 71 | \( ( 1 + 110852 T + 11683518662 T^{2} + 634743040311404 T^{3} + 35407535175826294175 T^{4} + \)\(12\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!64\)\( p^{5} T^{7} + 35407535175826294175 p^{10} T^{8} + 634743040311404 p^{15} T^{9} + 11683518662 p^{20} T^{10} + 110852 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 73 | \( 1 + 64232 T - 2926660864 T^{2} - 284906784229456 T^{3} + 2497920366784976909 T^{4} + \)\(80\!\cdots\!72\)\( T^{5} + \)\(73\!\cdots\!04\)\( T^{6} - \)\(23\!\cdots\!56\)\( T^{7} - \)\(82\!\cdots\!78\)\( T^{8} + \)\(39\!\cdots\!48\)\( T^{9} + \)\(28\!\cdots\!32\)\( T^{10} - \)\(23\!\cdots\!84\)\( T^{11} - \)\(63\!\cdots\!59\)\( T^{12} - \)\(23\!\cdots\!84\)\( p^{5} T^{13} + \)\(28\!\cdots\!32\)\( p^{10} T^{14} + \)\(39\!\cdots\!48\)\( p^{15} T^{15} - \)\(82\!\cdots\!78\)\( p^{20} T^{16} - \)\(23\!\cdots\!56\)\( p^{25} T^{17} + \)\(73\!\cdots\!04\)\( p^{30} T^{18} + \)\(80\!\cdots\!72\)\( p^{35} T^{19} + 2497920366784976909 p^{40} T^{20} - 284906784229456 p^{45} T^{21} - 2926660864 p^{50} T^{22} + 64232 p^{55} T^{23} + p^{60} T^{24} \) | |
| 79 | \( 1 + 111696 T - 4400478906 T^{2} - 536079012310208 T^{3} + 45823256450762664309 T^{4} + \)\(28\!\cdots\!64\)\( T^{5} - \)\(16\!\cdots\!22\)\( T^{6} - \)\(52\!\cdots\!04\)\( T^{7} + \)\(48\!\cdots\!30\)\( T^{8} + \)\(56\!\cdots\!72\)\( T^{9} - \)\(57\!\cdots\!26\)\( T^{10} - \)\(95\!\cdots\!76\)\( T^{11} + \)\(23\!\cdots\!21\)\( T^{12} - \)\(95\!\cdots\!76\)\( p^{5} T^{13} - \)\(57\!\cdots\!26\)\( p^{10} T^{14} + \)\(56\!\cdots\!72\)\( p^{15} T^{15} + \)\(48\!\cdots\!30\)\( p^{20} T^{16} - \)\(52\!\cdots\!04\)\( p^{25} T^{17} - \)\(16\!\cdots\!22\)\( p^{30} T^{18} + \)\(28\!\cdots\!64\)\( p^{35} T^{19} + 45823256450762664309 p^{40} T^{20} - 536079012310208 p^{45} T^{21} - 4400478906 p^{50} T^{22} + 111696 p^{55} T^{23} + p^{60} T^{24} \) | |
| 83 | \( ( 1 - 101128 T + 21476446850 T^{2} - 1732192887980152 T^{3} + \)\(20\!\cdots\!55\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(12\!\cdots\!00\)\( p^{5} T^{7} + \)\(20\!\cdots\!55\)\( p^{10} T^{8} - 1732192887980152 p^{15} T^{9} + 21476446850 p^{20} T^{10} - 101128 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 89 | \( 1 - 35012 T - 13927950220 T^{2} + 1727944897709464 T^{3} + 40710548335629414557 T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(92\!\cdots\!92\)\( T^{6} + \)\(21\!\cdots\!56\)\( T^{7} - \)\(61\!\cdots\!50\)\( T^{8} + \)\(58\!\cdots\!72\)\( T^{9} - \)\(20\!\cdots\!60\)\( T^{10} - \)\(23\!\cdots\!12\)\( T^{11} + \)\(35\!\cdots\!49\)\( T^{12} - \)\(23\!\cdots\!12\)\( p^{5} T^{13} - \)\(20\!\cdots\!60\)\( p^{10} T^{14} + \)\(58\!\cdots\!72\)\( p^{15} T^{15} - \)\(61\!\cdots\!50\)\( p^{20} T^{16} + \)\(21\!\cdots\!56\)\( p^{25} T^{17} + \)\(92\!\cdots\!92\)\( p^{30} T^{18} - \)\(15\!\cdots\!92\)\( p^{35} T^{19} + 40710548335629414557 p^{40} T^{20} + 1727944897709464 p^{45} T^{21} - 13927950220 p^{50} T^{22} - 35012 p^{55} T^{23} + p^{60} T^{24} \) | |
| 97 | \( ( 1 - 70952 T + 38177362064 T^{2} - 2926886015275816 T^{3} + \)\(65\!\cdots\!71\)\( T^{4} - \)\(49\!\cdots\!92\)\( T^{5} + \)\(69\!\cdots\!20\)\( T^{6} - \)\(49\!\cdots\!92\)\( p^{5} T^{7} + \)\(65\!\cdots\!71\)\( p^{10} T^{8} - 2926886015275816 p^{15} T^{9} + 38177362064 p^{20} T^{10} - 70952 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.61012847975474235171968712627, −3.36526867780575359864529230216, −3.17108339416318791120112167701, −3.09726449093255500350114947307, −3.00076554229018547765915176319, −2.99475833392667575633572341474, −2.95892122442717097740966620336, −2.89110194082055634373834967862, −2.72412054920661003792372208187, −2.32268495874871532115886534374, −2.11316279803754402268277645691, −1.86605140714313862321342904308, −1.79744034849521019901075873248, −1.70875169586454314612705475455, −1.58479163537371055861875116250, −1.42976967365618028285255605348, −1.42179358906136849326341339623, −1.20902909931353525503146644939, −0.75247912593173601622975537789, −0.63018685655651505440645101612, −0.62610657226854368535548144900, −0.43165391831811401161443841357, −0.39495248597323640326961296751, −0.38197842950764457984267666476, −0.15158850471808925230315670640, 0.15158850471808925230315670640, 0.38197842950764457984267666476, 0.39495248597323640326961296751, 0.43165391831811401161443841357, 0.62610657226854368535548144900, 0.63018685655651505440645101612, 0.75247912593173601622975537789, 1.20902909931353525503146644939, 1.42179358906136849326341339623, 1.42976967365618028285255605348, 1.58479163537371055861875116250, 1.70875169586454314612705475455, 1.79744034849521019901075873248, 1.86605140714313862321342904308, 2.11316279803754402268277645691, 2.32268495874871532115886534374, 2.72412054920661003792372208187, 2.89110194082055634373834967862, 2.95892122442717097740966620336, 2.99475833392667575633572341474, 3.00076554229018547765915176319, 3.09726449093255500350114947307, 3.17108339416318791120112167701, 3.36526867780575359864529230216, 3.61012847975474235171968712627
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.