Dirichlet series
| L(s) = 1 | − 190·3-s − 510·5-s − 4.64e3·7-s + 1.80e4·9-s − 5.70e4·11-s + 6.65e5·13-s + 9.69e4·15-s + 6.65e5·17-s + 8.82e5·21-s + 4.90e6·23-s − 4.92e6·25-s + 6.43e6·27-s + 1.16e7·31-s + 1.08e7·33-s + 2.36e6·35-s − 1.25e8·37-s − 1.26e8·39-s + 4.50e8·41-s − 3.68e8·43-s − 9.20e6·45-s + 4.69e8·47-s + 1.07e7·49-s − 1.26e8·51-s + 6.96e8·53-s + 2.90e7·55-s − 3.18e9·61-s − 8.38e7·63-s + ⋯ |
| L(s) = 1 | − 0.781·3-s − 0.163·5-s − 0.276·7-s + 0.305·9-s − 0.353·11-s + 1.79·13-s + 0.127·15-s + 0.468·17-s + 0.216·21-s + 0.762·23-s − 0.503·25-s + 0.448·27-s + 0.406·31-s + 0.276·33-s + 0.0451·35-s − 1.81·37-s − 1.40·39-s + 3.88·41-s − 2.50·43-s − 0.0498·45-s + 2.04·47-s + 0.0382·49-s − 0.366·51-s + 1.66·53-s + 0.0577·55-s − 3.77·61-s − 0.0844·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s+5)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42} \cdot 5^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.68912\times 10^{19}\) |
| Root analytic conductor: | \(5.04125\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} \cdot 5^{14} ,\ ( \ : [5]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.2122729444\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2122729444\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 102 p T + 207279 p^{2} T^{2} - 75165436 p^{4} T^{3} + 3515206613 p^{6} T^{4} - 287958079326 p^{8} T^{5} + 6149494935019 p^{12} T^{6} - 1402202695528 p^{17} T^{7} + 6149494935019 p^{22} T^{8} - 287958079326 p^{28} T^{9} + 3515206613 p^{36} T^{10} - 75165436 p^{44} T^{11} + 207279 p^{52} T^{12} + 102 p^{61} T^{13} + p^{70} T^{14} \) | |
| good | 3 | \( 1 + 190 T + 18050 T^{2} - 2146390 p T^{3} - 507570377 p^{2} T^{4} - 17649966572 p^{4} T^{5} - 2079589168580 p^{4} T^{6} - 32904262772780 p^{6} T^{7} + 164640533510629 p^{6} T^{8} + 660104148858326198 p^{7} T^{9} + \)\(10\!\cdots\!42\)\( p^{8} T^{10} + \)\(59\!\cdots\!90\)\( p^{12} T^{11} + \)\(29\!\cdots\!35\)\( p^{12} T^{12} - \)\(35\!\cdots\!04\)\( p^{13} T^{13} - \)\(42\!\cdots\!64\)\( p^{14} T^{14} - \)\(35\!\cdots\!04\)\( p^{23} T^{15} + \)\(29\!\cdots\!35\)\( p^{32} T^{16} + \)\(59\!\cdots\!90\)\( p^{42} T^{17} + \)\(10\!\cdots\!42\)\( p^{48} T^{18} + 660104148858326198 p^{57} T^{19} + 164640533510629 p^{66} T^{20} - 32904262772780 p^{76} T^{21} - 2079589168580 p^{84} T^{22} - 17649966572 p^{94} T^{23} - 507570377 p^{102} T^{24} - 2146390 p^{111} T^{25} + 18050 p^{120} T^{26} + 190 p^{130} T^{27} + p^{140} T^{28} \) |
| 7 | \( 1 + 4646 T + 10792658 T^{2} + 1554349651766 T^{3} - 93000566178886817 T^{4} + \)\(14\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!04\)\( T^{6} + \)\(29\!\cdots\!96\)\( p T^{7} + \)\(20\!\cdots\!13\)\( p^{2} T^{8} - \)\(19\!\cdots\!34\)\( p^{3} T^{9} + \)\(74\!\cdots\!22\)\( p^{4} T^{10} - \)\(18\!\cdots\!02\)\( p^{5} T^{11} - \)\(60\!\cdots\!53\)\( p^{6} T^{12} + \)\(17\!\cdots\!48\)\( p^{7} T^{13} + \)\(33\!\cdots\!84\)\( p^{8} T^{14} + \)\(17\!\cdots\!48\)\( p^{17} T^{15} - \)\(60\!\cdots\!53\)\( p^{26} T^{16} - \)\(18\!\cdots\!02\)\( p^{35} T^{17} + \)\(74\!\cdots\!22\)\( p^{44} T^{18} - \)\(19\!\cdots\!34\)\( p^{53} T^{19} + \)\(20\!\cdots\!13\)\( p^{62} T^{20} + \)\(29\!\cdots\!96\)\( p^{71} T^{21} + \)\(28\!\cdots\!04\)\( p^{80} T^{22} + \)\(14\!\cdots\!40\)\( p^{90} T^{23} - 93000566178886817 p^{100} T^{24} + 1554349651766 p^{110} T^{25} + 10792658 p^{120} T^{26} + 4646 p^{130} T^{27} + p^{140} T^{28} \) | |
| 11 | \( ( 1 + 28502 T + 101942997303 T^{2} + 7814589967399572 T^{3} + \)\(53\!\cdots\!01\)\( T^{4} + \)\(47\!\cdots\!74\)\( T^{5} + \)\(19\!\cdots\!23\)\( T^{6} + \)\(15\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!23\)\( p^{10} T^{8} + \)\(47\!\cdots\!74\)\( p^{20} T^{9} + \)\(53\!\cdots\!01\)\( p^{30} T^{10} + 7814589967399572 p^{40} T^{11} + 101942997303 p^{50} T^{12} + 28502 p^{60} T^{13} + p^{70} T^{14} )^{2} \) | |
| 13 | \( 1 - 665006 T + 221116490018 T^{2} - 54183802139411006 T^{3} + \)\(48\!\cdots\!43\)\( T^{4} - \)\(37\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!64\)\( p T^{6} - \)\(59\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!65\)\( T^{8} - \)\(10\!\cdots\!02\)\( T^{9} + \)\(43\!\cdots\!74\)\( T^{10} - \)\(19\!\cdots\!10\)\( T^{11} + \)\(82\!\cdots\!35\)\( T^{12} - \)\(19\!\cdots\!04\)\( p T^{13} + \)\(50\!\cdots\!56\)\( p^{2} T^{14} - \)\(19\!\cdots\!04\)\( p^{11} T^{15} + \)\(82\!\cdots\!35\)\( p^{20} T^{16} - \)\(19\!\cdots\!10\)\( p^{30} T^{17} + \)\(43\!\cdots\!74\)\( p^{40} T^{18} - \)\(10\!\cdots\!02\)\( p^{50} T^{19} + \)\(23\!\cdots\!65\)\( p^{60} T^{20} - \)\(59\!\cdots\!76\)\( p^{70} T^{21} + \)\(11\!\cdots\!64\)\( p^{81} T^{22} - \)\(37\!\cdots\!48\)\( p^{90} T^{23} + \)\(48\!\cdots\!43\)\( p^{100} T^{24} - 54183802139411006 p^{110} T^{25} + 221116490018 p^{120} T^{26} - 665006 p^{130} T^{27} + p^{140} T^{28} \) | |
| 17 | \( 1 - 665638 T + 221536973522 T^{2} - 2045093042954897926 T^{3} + \)\(57\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{5} - \)\(46\!\cdots\!48\)\( T^{6} + \)\(89\!\cdots\!28\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} - \)\(32\!\cdots\!30\)\( T^{9} + \)\(75\!\cdots\!30\)\( p T^{10} - \)\(62\!\cdots\!90\)\( T^{11} + \)\(18\!\cdots\!55\)\( T^{12} - \)\(55\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!60\)\( T^{14} - \)\(55\!\cdots\!40\)\( p^{10} T^{15} + \)\(18\!\cdots\!55\)\( p^{20} T^{16} - \)\(62\!\cdots\!90\)\( p^{30} T^{17} + \)\(75\!\cdots\!30\)\( p^{41} T^{18} - \)\(32\!\cdots\!30\)\( p^{50} T^{19} + \)\(21\!\cdots\!81\)\( p^{60} T^{20} + \)\(89\!\cdots\!28\)\( p^{70} T^{21} - \)\(46\!\cdots\!48\)\( p^{80} T^{22} + \)\(19\!\cdots\!16\)\( p^{90} T^{23} + \)\(57\!\cdots\!99\)\( p^{100} T^{24} - 2045093042954897926 p^{110} T^{25} + 221536973522 p^{120} T^{26} - 665638 p^{130} T^{27} + p^{140} T^{28} \) | |
| 19 | \( 1 - 63918751007966 T^{2} + \)\(19\!\cdots\!31\)\( T^{4} - \)\(38\!\cdots\!16\)\( T^{6} + \)\(55\!\cdots\!01\)\( T^{8} - \)\(59\!\cdots\!98\)\( T^{10} + \)\(26\!\cdots\!97\)\( p T^{12} - \)\(34\!\cdots\!88\)\( T^{14} + \)\(26\!\cdots\!97\)\( p^{21} T^{16} - \)\(59\!\cdots\!98\)\( p^{40} T^{18} + \)\(55\!\cdots\!01\)\( p^{60} T^{20} - \)\(38\!\cdots\!16\)\( p^{80} T^{22} + \)\(19\!\cdots\!31\)\( p^{100} T^{24} - 63918751007966 p^{120} T^{26} + p^{140} T^{28} \) | |
| 23 | \( 1 - 4904618 T + 12027638862962 T^{2} + \)\(48\!\cdots\!34\)\( T^{3} + \)\(36\!\cdots\!39\)\( T^{4} - \)\(41\!\cdots\!32\)\( T^{5} + \)\(27\!\cdots\!36\)\( T^{6} + \)\(18\!\cdots\!72\)\( T^{7} - \)\(48\!\cdots\!51\)\( T^{8} - \)\(93\!\cdots\!62\)\( T^{9} + \)\(12\!\cdots\!34\)\( T^{10} + \)\(18\!\cdots\!18\)\( T^{11} - \)\(14\!\cdots\!85\)\( T^{12} - \)\(58\!\cdots\!20\)\( T^{13} + \)\(26\!\cdots\!28\)\( T^{14} - \)\(58\!\cdots\!20\)\( p^{10} T^{15} - \)\(14\!\cdots\!85\)\( p^{20} T^{16} + \)\(18\!\cdots\!18\)\( p^{30} T^{17} + \)\(12\!\cdots\!34\)\( p^{40} T^{18} - \)\(93\!\cdots\!62\)\( p^{50} T^{19} - \)\(48\!\cdots\!51\)\( p^{60} T^{20} + \)\(18\!\cdots\!72\)\( p^{70} T^{21} + \)\(27\!\cdots\!36\)\( p^{80} T^{22} - \)\(41\!\cdots\!32\)\( p^{90} T^{23} + \)\(36\!\cdots\!39\)\( p^{100} T^{24} + \)\(48\!\cdots\!34\)\( p^{110} T^{25} + 12027638862962 p^{120} T^{26} - 4904618 p^{130} T^{27} + p^{140} T^{28} \) | |
| 29 | \( 1 - 3287498286051902 T^{2} + \)\(54\!\cdots\!23\)\( T^{4} - \)\(59\!\cdots\!72\)\( T^{6} + \)\(47\!\cdots\!61\)\( T^{8} - \)\(30\!\cdots\!66\)\( T^{10} + \)\(16\!\cdots\!27\)\( T^{12} - \)\(73\!\cdots\!88\)\( T^{14} + \)\(16\!\cdots\!27\)\( p^{20} T^{16} - \)\(30\!\cdots\!66\)\( p^{40} T^{18} + \)\(47\!\cdots\!61\)\( p^{60} T^{20} - \)\(59\!\cdots\!72\)\( p^{80} T^{22} + \)\(54\!\cdots\!23\)\( p^{100} T^{24} - 3287498286051902 p^{120} T^{26} + p^{140} T^{28} \) | |
| 31 | \( ( 1 - 5812074 T + 2481756326114991 T^{2} + \)\(92\!\cdots\!76\)\( T^{3} + \)\(36\!\cdots\!41\)\( T^{4} + \)\(10\!\cdots\!70\)\( T^{5} + \)\(38\!\cdots\!75\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(38\!\cdots\!75\)\( p^{10} T^{8} + \)\(10\!\cdots\!70\)\( p^{20} T^{9} + \)\(36\!\cdots\!41\)\( p^{30} T^{10} + \)\(92\!\cdots\!76\)\( p^{40} T^{11} + 2481756326114991 p^{50} T^{12} - 5812074 p^{60} T^{13} + p^{70} T^{14} )^{2} \) | |
| 37 | \( 1 + 125728506 T + 7903828610496018 T^{2} + \)\(63\!\cdots\!26\)\( T^{3} + \)\(59\!\cdots\!03\)\( T^{4} + \)\(51\!\cdots\!16\)\( T^{5} + \)\(36\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!73\)\( T^{8} + \)\(17\!\cdots\!46\)\( T^{9} + \)\(12\!\cdots\!50\)\( T^{10} + \)\(61\!\cdots\!38\)\( T^{11} + \)\(32\!\cdots\!79\)\( T^{12} + \)\(28\!\cdots\!84\)\( T^{13} + \)\(25\!\cdots\!16\)\( T^{14} + \)\(28\!\cdots\!84\)\( p^{10} T^{15} + \)\(32\!\cdots\!79\)\( p^{20} T^{16} + \)\(61\!\cdots\!38\)\( p^{30} T^{17} + \)\(12\!\cdots\!50\)\( p^{40} T^{18} + \)\(17\!\cdots\!46\)\( p^{50} T^{19} + \)\(15\!\cdots\!73\)\( p^{60} T^{20} + \)\(17\!\cdots\!00\)\( p^{70} T^{21} + \)\(36\!\cdots\!80\)\( p^{80} T^{22} + \)\(51\!\cdots\!16\)\( p^{90} T^{23} + \)\(59\!\cdots\!03\)\( p^{100} T^{24} + \)\(63\!\cdots\!26\)\( p^{110} T^{25} + 7903828610496018 p^{120} T^{26} + 125728506 p^{130} T^{27} + p^{140} T^{28} \) | |
| 41 | \( ( 1 - 225218138 T + 39563408112962623 T^{2} - \)\(21\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} - \)\(70\!\cdots\!62\)\( T^{5} + \)\(61\!\cdots\!51\)\( T^{6} - \)\(54\!\cdots\!96\)\( T^{7} + \)\(61\!\cdots\!51\)\( p^{10} T^{8} - \)\(70\!\cdots\!62\)\( p^{20} T^{9} + \)\(21\!\cdots\!61\)\( p^{30} T^{10} - \)\(21\!\cdots\!88\)\( p^{40} T^{11} + 39563408112962623 p^{50} T^{12} - 225218138 p^{60} T^{13} + p^{70} T^{14} )^{2} \) | |
| 43 | \( 1 + 368542126 T + 67911649318299938 T^{2} + \)\(57\!\cdots\!66\)\( T^{3} - \)\(24\!\cdots\!17\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{5} - \)\(14\!\cdots\!36\)\( T^{6} - \)\(89\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!37\)\( T^{8} + \)\(82\!\cdots\!38\)\( T^{9} + \)\(18\!\cdots\!22\)\( T^{10} + \)\(26\!\cdots\!06\)\( T^{11} + \)\(10\!\cdots\!63\)\( T^{12} - \)\(34\!\cdots\!36\)\( T^{13} - \)\(81\!\cdots\!16\)\( T^{14} - \)\(34\!\cdots\!36\)\( p^{10} T^{15} + \)\(10\!\cdots\!63\)\( p^{20} T^{16} + \)\(26\!\cdots\!06\)\( p^{30} T^{17} + \)\(18\!\cdots\!22\)\( p^{40} T^{18} + \)\(82\!\cdots\!38\)\( p^{50} T^{19} + \)\(17\!\cdots\!37\)\( p^{60} T^{20} - \)\(89\!\cdots\!28\)\( p^{70} T^{21} - \)\(14\!\cdots\!36\)\( p^{80} T^{22} - \)\(13\!\cdots\!60\)\( p^{90} T^{23} - \)\(24\!\cdots\!17\)\( p^{100} T^{24} + \)\(57\!\cdots\!66\)\( p^{110} T^{25} + 67911649318299938 p^{120} T^{26} + 368542126 p^{130} T^{27} + p^{140} T^{28} \) | |
| 47 | \( 1 - 469365626 T + 110152045435185938 T^{2} - \)\(38\!\cdots\!66\)\( T^{3} - \)\(31\!\cdots\!77\)\( T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} - \)\(32\!\cdots\!08\)\( T^{7} + \)\(65\!\cdots\!89\)\( T^{8} + \)\(82\!\cdots\!66\)\( T^{9} - \)\(68\!\cdots\!26\)\( T^{10} + \)\(15\!\cdots\!74\)\( T^{11} - \)\(14\!\cdots\!57\)\( T^{12} - \)\(76\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!16\)\( T^{14} - \)\(76\!\cdots\!20\)\( p^{10} T^{15} - \)\(14\!\cdots\!57\)\( p^{20} T^{16} + \)\(15\!\cdots\!74\)\( p^{30} T^{17} - \)\(68\!\cdots\!26\)\( p^{40} T^{18} + \)\(82\!\cdots\!66\)\( p^{50} T^{19} + \)\(65\!\cdots\!89\)\( p^{60} T^{20} - \)\(32\!\cdots\!08\)\( p^{70} T^{21} + \)\(12\!\cdots\!76\)\( p^{80} T^{22} + \)\(71\!\cdots\!28\)\( p^{90} T^{23} - \)\(31\!\cdots\!77\)\( p^{100} T^{24} - \)\(38\!\cdots\!66\)\( p^{110} T^{25} + 110152045435185938 p^{120} T^{26} - 469365626 p^{130} T^{27} + p^{140} T^{28} \) | |
| 53 | \( 1 - 696851622 T + 242801091542015442 T^{2} + \)\(75\!\cdots\!06\)\( T^{3} - \)\(95\!\cdots\!21\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(92\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!25\)\( T^{8} + \)\(11\!\cdots\!66\)\( T^{9} - \)\(32\!\cdots\!86\)\( T^{10} - \)\(12\!\cdots\!78\)\( T^{11} + \)\(81\!\cdots\!19\)\( T^{12} + \)\(73\!\cdots\!72\)\( T^{13} - \)\(15\!\cdots\!68\)\( T^{14} + \)\(73\!\cdots\!72\)\( p^{10} T^{15} + \)\(81\!\cdots\!19\)\( p^{20} T^{16} - \)\(12\!\cdots\!78\)\( p^{30} T^{17} - \)\(32\!\cdots\!86\)\( p^{40} T^{18} + \)\(11\!\cdots\!66\)\( p^{50} T^{19} + \)\(10\!\cdots\!25\)\( p^{60} T^{20} - \)\(92\!\cdots\!80\)\( p^{70} T^{21} + \)\(13\!\cdots\!04\)\( p^{80} T^{22} + \)\(18\!\cdots\!68\)\( p^{90} T^{23} - \)\(95\!\cdots\!21\)\( p^{100} T^{24} + \)\(75\!\cdots\!06\)\( p^{110} T^{25} + 242801091542015442 p^{120} T^{26} - 696851622 p^{130} T^{27} + p^{140} T^{28} \) | |
| 59 | \( 1 - 3217927897027278846 T^{2} + \)\(48\!\cdots\!31\)\( T^{4} - \)\(45\!\cdots\!56\)\( T^{6} + \)\(30\!\cdots\!81\)\( T^{8} - \)\(15\!\cdots\!46\)\( T^{10} + \)\(63\!\cdots\!91\)\( T^{12} - \)\(28\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!91\)\( p^{20} T^{16} - \)\(15\!\cdots\!46\)\( p^{40} T^{18} + \)\(30\!\cdots\!81\)\( p^{60} T^{20} - \)\(45\!\cdots\!56\)\( p^{80} T^{22} + \)\(48\!\cdots\!31\)\( p^{100} T^{24} - 3217927897027278846 p^{120} T^{26} + p^{140} T^{28} \) | |
| 61 | \( ( 1 + 1593489294 T + 3715555055655916071 T^{2} + \)\(45\!\cdots\!24\)\( T^{3} + \)\(66\!\cdots\!41\)\( T^{4} + \)\(66\!\cdots\!50\)\( T^{5} + \)\(72\!\cdots\!75\)\( T^{6} + \)\(58\!\cdots\!40\)\( T^{7} + \)\(72\!\cdots\!75\)\( p^{10} T^{8} + \)\(66\!\cdots\!50\)\( p^{20} T^{9} + \)\(66\!\cdots\!41\)\( p^{30} T^{10} + \)\(45\!\cdots\!24\)\( p^{40} T^{11} + 3715555055655916071 p^{50} T^{12} + 1593489294 p^{60} T^{13} + p^{70} T^{14} )^{2} \) | |
| 67 | \( 1 + 1880150718 T + 1767483361197957762 T^{2} + \)\(36\!\cdots\!06\)\( T^{3} + \)\(46\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!88\)\( T^{6} + \)\(30\!\cdots\!36\)\( T^{7} + \)\(53\!\cdots\!53\)\( T^{8} + \)\(62\!\cdots\!38\)\( T^{9} + \)\(10\!\cdots\!30\)\( T^{10} + \)\(14\!\cdots\!70\)\( T^{11} + \)\(45\!\cdots\!27\)\( p^{2} T^{12} + \)\(49\!\cdots\!52\)\( p T^{13} + \)\(40\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!52\)\( p^{11} T^{15} + \)\(45\!\cdots\!27\)\( p^{22} T^{16} + \)\(14\!\cdots\!70\)\( p^{30} T^{17} + \)\(10\!\cdots\!30\)\( p^{40} T^{18} + \)\(62\!\cdots\!38\)\( p^{50} T^{19} + \)\(53\!\cdots\!53\)\( p^{60} T^{20} + \)\(30\!\cdots\!36\)\( p^{70} T^{21} + \)\(20\!\cdots\!88\)\( p^{80} T^{22} + \)\(11\!\cdots\!96\)\( p^{90} T^{23} + \)\(46\!\cdots\!59\)\( p^{100} T^{24} + \)\(36\!\cdots\!06\)\( p^{110} T^{25} + 1767483361197957762 p^{120} T^{26} + 1880150718 p^{130} T^{27} + p^{140} T^{28} \) | |
| 71 | \( ( 1 - 2357367930 T + 6908930409171076287 T^{2} + \)\(14\!\cdots\!20\)\( T^{3} - \)\(25\!\cdots\!39\)\( T^{4} + \)\(42\!\cdots\!62\)\( T^{5} + \)\(22\!\cdots\!15\)\( T^{6} + \)\(46\!\cdots\!24\)\( T^{7} + \)\(22\!\cdots\!15\)\( p^{10} T^{8} + \)\(42\!\cdots\!62\)\( p^{20} T^{9} - \)\(25\!\cdots\!39\)\( p^{30} T^{10} + \)\(14\!\cdots\!20\)\( p^{40} T^{11} + 6908930409171076287 p^{50} T^{12} - 2357367930 p^{60} T^{13} + p^{70} T^{14} )^{2} \) | |
| 73 | \( 1 + 3104937922 T + 4820319749736839042 T^{2} + \)\(78\!\cdots\!94\)\( T^{3} - \)\(20\!\cdots\!21\)\( T^{4} - \)\(86\!\cdots\!12\)\( T^{5} - \)\(16\!\cdots\!64\)\( T^{6} - \)\(26\!\cdots\!28\)\( T^{7} - \)\(71\!\cdots\!11\)\( T^{8} + \)\(26\!\cdots\!18\)\( T^{9} + \)\(12\!\cdots\!54\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{11} + \)\(36\!\cdots\!75\)\( T^{12} + \)\(88\!\cdots\!60\)\( T^{13} + \)\(37\!\cdots\!48\)\( T^{14} + \)\(88\!\cdots\!60\)\( p^{10} T^{15} + \)\(36\!\cdots\!75\)\( p^{20} T^{16} + \)\(15\!\cdots\!38\)\( p^{30} T^{17} + \)\(12\!\cdots\!54\)\( p^{40} T^{18} + \)\(26\!\cdots\!18\)\( p^{50} T^{19} - \)\(71\!\cdots\!11\)\( p^{60} T^{20} - \)\(26\!\cdots\!28\)\( p^{70} T^{21} - \)\(16\!\cdots\!64\)\( p^{80} T^{22} - \)\(86\!\cdots\!12\)\( p^{90} T^{23} - \)\(20\!\cdots\!21\)\( p^{100} T^{24} + \)\(78\!\cdots\!94\)\( p^{110} T^{25} + 4820319749736839042 p^{120} T^{26} + 3104937922 p^{130} T^{27} + p^{140} T^{28} \) | |
| 79 | \( 1 - 46181029418221446734 T^{2} + \)\(13\!\cdots\!51\)\( T^{4} - \)\(27\!\cdots\!84\)\( T^{6} + \)\(44\!\cdots\!41\)\( T^{8} - \)\(59\!\cdots\!58\)\( T^{10} + \)\(69\!\cdots\!07\)\( T^{12} - \)\(70\!\cdots\!48\)\( T^{14} + \)\(69\!\cdots\!07\)\( p^{20} T^{16} - \)\(59\!\cdots\!58\)\( p^{40} T^{18} + \)\(44\!\cdots\!41\)\( p^{60} T^{20} - \)\(27\!\cdots\!84\)\( p^{80} T^{22} + \)\(13\!\cdots\!51\)\( p^{100} T^{24} - 46181029418221446734 p^{120} T^{26} + p^{140} T^{28} \) | |
| 83 | \( 1 - 1333655538 T + 889318547019034722 T^{2} + \)\(28\!\cdots\!94\)\( T^{3} + \)\(18\!\cdots\!79\)\( T^{4} - \)\(14\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!56\)\( T^{7} - \)\(77\!\cdots\!67\)\( T^{8} + \)\(69\!\cdots\!42\)\( T^{9} + \)\(19\!\cdots\!10\)\( T^{10} - \)\(41\!\cdots\!30\)\( T^{11} + \)\(80\!\cdots\!03\)\( T^{12} + \)\(91\!\cdots\!76\)\( T^{13} - \)\(12\!\cdots\!52\)\( T^{14} + \)\(91\!\cdots\!76\)\( p^{10} T^{15} + \)\(80\!\cdots\!03\)\( p^{20} T^{16} - \)\(41\!\cdots\!30\)\( p^{30} T^{17} + \)\(19\!\cdots\!10\)\( p^{40} T^{18} + \)\(69\!\cdots\!42\)\( p^{50} T^{19} - \)\(77\!\cdots\!67\)\( p^{60} T^{20} - \)\(13\!\cdots\!56\)\( p^{70} T^{21} + \)\(21\!\cdots\!28\)\( p^{80} T^{22} - \)\(14\!\cdots\!96\)\( p^{90} T^{23} + \)\(18\!\cdots\!79\)\( p^{100} T^{24} + \)\(28\!\cdots\!94\)\( p^{110} T^{25} + 889318547019034722 p^{120} T^{26} - 1333655538 p^{130} T^{27} + p^{140} T^{28} \) | |
| 89 | \( 1 - \)\(22\!\cdots\!06\)\( T^{2} + \)\(27\!\cdots\!51\)\( T^{4} - \)\(22\!\cdots\!56\)\( T^{6} + \)\(13\!\cdots\!01\)\( T^{8} - \)\(67\!\cdots\!98\)\( T^{10} + \)\(27\!\cdots\!63\)\( T^{12} - \)\(94\!\cdots\!48\)\( T^{14} + \)\(27\!\cdots\!63\)\( p^{20} T^{16} - \)\(67\!\cdots\!98\)\( p^{40} T^{18} + \)\(13\!\cdots\!01\)\( p^{60} T^{20} - \)\(22\!\cdots\!56\)\( p^{80} T^{22} + \)\(27\!\cdots\!51\)\( p^{100} T^{24} - \)\(22\!\cdots\!06\)\( p^{120} T^{26} + p^{140} T^{28} \) | |
| 97 | \( 1 - 1599867854 T + 1279788575131282658 T^{2} - \)\(89\!\cdots\!34\)\( T^{3} - \)\(70\!\cdots\!77\)\( T^{4} + \)\(43\!\cdots\!72\)\( T^{5} + \)\(33\!\cdots\!56\)\( T^{6} + \)\(43\!\cdots\!48\)\( T^{7} - \)\(40\!\cdots\!11\)\( T^{8} - \)\(22\!\cdots\!66\)\( T^{9} - \)\(15\!\cdots\!26\)\( T^{10} + \)\(38\!\cdots\!26\)\( T^{11} + \)\(27\!\cdots\!23\)\( T^{12} - \)\(25\!\cdots\!80\)\( T^{13} - \)\(18\!\cdots\!84\)\( T^{14} - \)\(25\!\cdots\!80\)\( p^{10} T^{15} + \)\(27\!\cdots\!23\)\( p^{20} T^{16} + \)\(38\!\cdots\!26\)\( p^{30} T^{17} - \)\(15\!\cdots\!26\)\( p^{40} T^{18} - \)\(22\!\cdots\!66\)\( p^{50} T^{19} - \)\(40\!\cdots\!11\)\( p^{60} T^{20} + \)\(43\!\cdots\!48\)\( p^{70} T^{21} + \)\(33\!\cdots\!56\)\( p^{80} T^{22} + \)\(43\!\cdots\!72\)\( p^{90} T^{23} - \)\(70\!\cdots\!77\)\( p^{100} T^{24} - \)\(89\!\cdots\!34\)\( p^{110} T^{25} + 1279788575131282658 p^{120} T^{26} - 1599867854 p^{130} T^{27} + p^{140} T^{28} \) | |
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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
−3.43610520112882588047658985048, −3.21847758617002068105023955902, −2.80528393040538308708661858914, −2.79726908547060110802004860167, −2.68244284549349099180295182706, −2.61966986529246889578961139378, −2.51097566955553794052720796406, −2.48413433200695572343131006582, −2.47310233207940923469427206397, −2.22040914902043674512960172573, −1.81767909109466009042698133662, −1.76756943626780709843389649202, −1.67277196808112926330100403934, −1.43184537102217555252103986963, −1.36953893881099089656172476699, −1.21629837200293662671091974072, −1.08771884470345986057369515738, −1.08207984757826769205614783479, −1.07342197616462623604026981465, −0.911374991603251723729118031831, −0.60884459360207501554769108612, −0.41137040621053502548544191029, −0.24232714295252463985246357315, −0.11013346459999619035402628707, −0.06693518776175911599924317451, 0.06693518776175911599924317451, 0.11013346459999619035402628707, 0.24232714295252463985246357315, 0.41137040621053502548544191029, 0.60884459360207501554769108612, 0.911374991603251723729118031831, 1.07342197616462623604026981465, 1.08207984757826769205614783479, 1.08771884470345986057369515738, 1.21629837200293662671091974072, 1.36953893881099089656172476699, 1.43184537102217555252103986963, 1.67277196808112926330100403934, 1.76756943626780709843389649202, 1.81767909109466009042698133662, 2.22040914902043674512960172573, 2.47310233207940923469427206397, 2.48413433200695572343131006582, 2.51097566955553794052720796406, 2.61966986529246889578961139378, 2.68244284549349099180295182706, 2.79726908547060110802004860167, 2.80528393040538308708661858914, 3.21847758617002068105023955902, 3.43610520112882588047658985048
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.