Dirichlet series
| L(s) = 1 | + 5.37e3·5-s − 3.73e3·7-s + 4.65e3·11-s + 4.74e4·17-s − 9.59e6·19-s + 1.40e6·23-s − 1.41e7·25-s − 7.30e7·29-s − 2.76e7·31-s − 2.01e7·35-s − 3.15e7·37-s + 5.19e7·43-s + 3.90e7·47-s − 3.38e8·49-s − 4.82e8·53-s + 2.50e7·55-s − 3.36e8·59-s + 5.86e8·61-s + 2.55e8·67-s − 1.87e9·71-s − 4.37e9·73-s − 1.74e7·77-s + 5.06e8·79-s + 2.55e8·85-s + 1.40e10·89-s − 5.16e10·95-s − 3.90e9·101-s + ⋯ |
| L(s) = 1 | + 1.72·5-s − 0.222·7-s + 0.0289·11-s + 0.0334·17-s − 3.87·19-s + 0.218·23-s − 1.44·25-s − 3.56·29-s − 0.966·31-s − 0.382·35-s − 0.455·37-s + 0.353·43-s + 0.170·47-s − 1.19·49-s − 1.15·53-s + 0.0497·55-s − 0.470·59-s + 0.693·61-s + 0.189·67-s − 1.03·71-s − 2.11·73-s − 0.00643·77-s + 0.164·79-s + 0.0575·85-s + 2.51·89-s − 6.67·95-s − 0.371·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.83806\times 10^{26}\) |
| Root analytic conductor: | \(12.6534\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [5]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.0001385772756\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0001385772756\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 + 534 p T + 7192245 p^{2} T^{2} + 550631170 p^{5} T^{3} + 25156512666 p^{8} T^{4} + 145227875490 p^{12} T^{5} + 2428995389181 p^{16} T^{6} + 145227875490 p^{22} T^{7} + 25156512666 p^{28} T^{8} + 550631170 p^{35} T^{9} + 7192245 p^{42} T^{10} + 534 p^{51} T^{11} + p^{60} T^{12} \) | |
| good | 5 | \( 1 - 5379 T + 1722588 p^{2} T^{2} - 179767002987 T^{3} + 770486347207434 T^{4} - 534539183857337211 p T^{5} + \)\(18\!\cdots\!18\)\( p T^{6} - \)\(22\!\cdots\!73\)\( p^{3} T^{7} + \)\(40\!\cdots\!22\)\( p^{2} T^{8} - \)\(22\!\cdots\!53\)\( p^{3} T^{9} + \)\(77\!\cdots\!28\)\( p^{3} T^{10} - \)\(42\!\cdots\!89\)\( p^{4} T^{11} + \)\(13\!\cdots\!94\)\( p^{4} T^{12} - \)\(42\!\cdots\!89\)\( p^{14} T^{13} + \)\(77\!\cdots\!28\)\( p^{23} T^{14} - \)\(22\!\cdots\!53\)\( p^{33} T^{15} + \)\(40\!\cdots\!22\)\( p^{42} T^{16} - \)\(22\!\cdots\!73\)\( p^{53} T^{17} + \)\(18\!\cdots\!18\)\( p^{61} T^{18} - 534539183857337211 p^{71} T^{19} + 770486347207434 p^{80} T^{20} - 179767002987 p^{90} T^{21} + 1722588 p^{102} T^{22} - 5379 p^{110} T^{23} + p^{120} T^{24} \) |
| 11 | \( 1 - 4659 T - 90364425090 T^{2} - 16388479069347225 T^{3} + \)\(52\!\cdots\!64\)\( T^{4} + \)\(13\!\cdots\!49\)\( T^{5} - \)\(67\!\cdots\!38\)\( T^{6} - \)\(75\!\cdots\!89\)\( T^{7} - \)\(45\!\cdots\!80\)\( T^{8} + \)\(20\!\cdots\!05\)\( T^{9} + \)\(39\!\cdots\!46\)\( T^{10} - \)\(26\!\cdots\!41\)\( T^{11} - \)\(12\!\cdots\!30\)\( T^{12} - \)\(26\!\cdots\!41\)\( p^{10} T^{13} + \)\(39\!\cdots\!46\)\( p^{20} T^{14} + \)\(20\!\cdots\!05\)\( p^{30} T^{15} - \)\(45\!\cdots\!80\)\( p^{40} T^{16} - \)\(75\!\cdots\!89\)\( p^{50} T^{17} - \)\(67\!\cdots\!38\)\( p^{60} T^{18} + \)\(13\!\cdots\!49\)\( p^{70} T^{19} + \)\(52\!\cdots\!64\)\( p^{80} T^{20} - 16388479069347225 p^{90} T^{21} - 90364425090 p^{100} T^{22} - 4659 p^{110} T^{23} + p^{120} T^{24} \) | |
| 13 | \( 1 - 692093911827 T^{2} + \)\(27\!\cdots\!07\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!73\)\( p T^{8} - \)\(32\!\cdots\!25\)\( T^{10} + \)\(49\!\cdots\!86\)\( T^{12} - \)\(32\!\cdots\!25\)\( p^{20} T^{14} + \)\(13\!\cdots\!73\)\( p^{41} T^{16} - \)\(79\!\cdots\!20\)\( p^{60} T^{18} + \)\(27\!\cdots\!07\)\( p^{80} T^{20} - 692093911827 p^{100} T^{22} + p^{120} T^{24} \) | |
| 17 | \( 1 - 47478 T + 5201395633662 T^{2} - 246916187551184652 T^{3} + \)\(14\!\cdots\!53\)\( T^{4} - \)\(90\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!98\)\( T^{6} + \)\(86\!\cdots\!90\)\( T^{7} + \)\(62\!\cdots\!90\)\( T^{8} + \)\(52\!\cdots\!30\)\( T^{9} + \)\(77\!\cdots\!98\)\( T^{10} + \)\(12\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!85\)\( T^{12} + \)\(12\!\cdots\!72\)\( p^{10} T^{13} + \)\(77\!\cdots\!98\)\( p^{20} T^{14} + \)\(52\!\cdots\!30\)\( p^{30} T^{15} + \)\(62\!\cdots\!90\)\( p^{40} T^{16} + \)\(86\!\cdots\!90\)\( p^{50} T^{17} + \)\(35\!\cdots\!98\)\( p^{60} T^{18} - \)\(90\!\cdots\!80\)\( p^{70} T^{19} + \)\(14\!\cdots\!53\)\( p^{80} T^{20} - 246916187551184652 p^{90} T^{21} + 5201395633662 p^{100} T^{22} - 47478 p^{110} T^{23} + p^{120} T^{24} \) | |
| 19 | \( 1 + 9596979 T + 60686649719520 T^{2} + \)\(28\!\cdots\!67\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} + \)\(37\!\cdots\!59\)\( T^{5} + \)\(10\!\cdots\!30\)\( T^{6} + \)\(28\!\cdots\!81\)\( T^{7} + \)\(66\!\cdots\!18\)\( T^{8} + \)\(14\!\cdots\!53\)\( T^{9} + \)\(30\!\cdots\!44\)\( T^{10} + \)\(63\!\cdots\!61\)\( T^{11} + \)\(14\!\cdots\!78\)\( T^{12} + \)\(63\!\cdots\!61\)\( p^{10} T^{13} + \)\(30\!\cdots\!44\)\( p^{20} T^{14} + \)\(14\!\cdots\!53\)\( p^{30} T^{15} + \)\(66\!\cdots\!18\)\( p^{40} T^{16} + \)\(28\!\cdots\!81\)\( p^{50} T^{17} + \)\(10\!\cdots\!30\)\( p^{60} T^{18} + \)\(37\!\cdots\!59\)\( p^{70} T^{19} + \)\(11\!\cdots\!98\)\( p^{80} T^{20} + \)\(28\!\cdots\!67\)\( p^{90} T^{21} + 60686649719520 p^{100} T^{22} + 9596979 p^{110} T^{23} + p^{120} T^{24} \) | |
| 23 | \( 1 - 1406448 T - 209986742464998 T^{2} - 57930109728707216448 T^{3} + \)\(25\!\cdots\!49\)\( T^{4} + \)\(11\!\cdots\!32\)\( p T^{5} - \)\(20\!\cdots\!98\)\( T^{6} - \)\(27\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!02\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} - \)\(63\!\cdots\!62\)\( T^{10} - \)\(26\!\cdots\!56\)\( T^{11} + \)\(27\!\cdots\!01\)\( T^{12} - \)\(26\!\cdots\!56\)\( p^{10} T^{13} - \)\(63\!\cdots\!62\)\( p^{20} T^{14} + \)\(14\!\cdots\!80\)\( p^{30} T^{15} + \)\(12\!\cdots\!02\)\( p^{40} T^{16} - \)\(27\!\cdots\!28\)\( p^{50} T^{17} - \)\(20\!\cdots\!98\)\( p^{60} T^{18} + \)\(11\!\cdots\!32\)\( p^{71} T^{19} + \)\(25\!\cdots\!49\)\( p^{80} T^{20} - 57930109728707216448 p^{90} T^{21} - 209986742464998 p^{100} T^{22} - 1406448 p^{110} T^{23} + p^{120} T^{24} \) | |
| 29 | \( ( 1 + 36520113 T + 1791031479972513 T^{2} + \)\(42\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!03\)\( T^{4} + \)\(28\!\cdots\!91\)\( T^{5} + \)\(71\!\cdots\!46\)\( T^{6} + \)\(28\!\cdots\!91\)\( p^{10} T^{7} + \)\(13\!\cdots\!03\)\( p^{20} T^{8} + \)\(42\!\cdots\!08\)\( p^{30} T^{9} + 1791031479972513 p^{40} T^{10} + 36520113 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 31 | \( 1 + 27662112 T + 4160376651638373 T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(92\!\cdots\!55\)\( T^{4} + \)\(20\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(23\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!85\)\( T^{8} + \)\(20\!\cdots\!32\)\( T^{9} + \)\(13\!\cdots\!15\)\( T^{10} + \)\(14\!\cdots\!72\)\( T^{11} + \)\(10\!\cdots\!78\)\( T^{12} + \)\(14\!\cdots\!72\)\( p^{10} T^{13} + \)\(13\!\cdots\!15\)\( p^{20} T^{14} + \)\(20\!\cdots\!32\)\( p^{30} T^{15} + \)\(14\!\cdots\!85\)\( p^{40} T^{16} + \)\(23\!\cdots\!68\)\( p^{50} T^{17} + \)\(13\!\cdots\!24\)\( p^{60} T^{18} + \)\(20\!\cdots\!16\)\( p^{70} T^{19} + \)\(92\!\cdots\!55\)\( p^{80} T^{20} + \)\(10\!\cdots\!00\)\( p^{90} T^{21} + 4160376651638373 p^{100} T^{22} + 27662112 p^{110} T^{23} + p^{120} T^{24} \) | |
| 37 | \( 1 + 31582365 T - 17799960008820486 T^{2} - \)\(22\!\cdots\!29\)\( p T^{3} + \)\(14\!\cdots\!88\)\( T^{4} + \)\(82\!\cdots\!61\)\( T^{5} - \)\(88\!\cdots\!94\)\( T^{6} - \)\(38\!\cdots\!69\)\( T^{7} + \)\(54\!\cdots\!44\)\( T^{8} + \)\(75\!\cdots\!05\)\( T^{9} - \)\(36\!\cdots\!30\)\( T^{10} - \)\(31\!\cdots\!97\)\( T^{11} + \)\(20\!\cdots\!90\)\( T^{12} - \)\(31\!\cdots\!97\)\( p^{10} T^{13} - \)\(36\!\cdots\!30\)\( p^{20} T^{14} + \)\(75\!\cdots\!05\)\( p^{30} T^{15} + \)\(54\!\cdots\!44\)\( p^{40} T^{16} - \)\(38\!\cdots\!69\)\( p^{50} T^{17} - \)\(88\!\cdots\!94\)\( p^{60} T^{18} + \)\(82\!\cdots\!61\)\( p^{70} T^{19} + \)\(14\!\cdots\!88\)\( p^{80} T^{20} - \)\(22\!\cdots\!29\)\( p^{91} T^{21} - 17799960008820486 p^{100} T^{22} + 31582365 p^{110} T^{23} + p^{120} T^{24} \) | |
| 41 | \( 1 - 132426866136187236 T^{2} + \)\(20\!\cdots\!26\)\( p T^{4} - \)\(32\!\cdots\!88\)\( T^{6} + \)\(90\!\cdots\!79\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!08\)\( T^{12} - \)\(18\!\cdots\!84\)\( p^{20} T^{14} + \)\(90\!\cdots\!79\)\( p^{40} T^{16} - \)\(32\!\cdots\!88\)\( p^{60} T^{18} + \)\(20\!\cdots\!26\)\( p^{81} T^{20} - 132426866136187236 p^{100} T^{22} + p^{120} T^{24} \) | |
| 43 | \( ( 1 - 25983321 T + 10243827272600859 T^{2} + \)\(30\!\cdots\!92\)\( T^{3} - \)\(73\!\cdots\!31\)\( T^{4} + \)\(28\!\cdots\!69\)\( T^{5} + \)\(91\!\cdots\!62\)\( T^{6} + \)\(28\!\cdots\!69\)\( p^{10} T^{7} - \)\(73\!\cdots\!31\)\( p^{20} T^{8} + \)\(30\!\cdots\!92\)\( p^{30} T^{9} + 10243827272600859 p^{40} T^{10} - 25983321 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 47 | \( 1 - 39022350 T + 208293091597001526 T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!53\)\( T^{4} + \)\(16\!\cdots\!80\)\( T^{5} + \)\(17\!\cdots\!78\)\( T^{6} + \)\(15\!\cdots\!90\)\( T^{7} + \)\(81\!\cdots\!46\)\( T^{8} + \)\(21\!\cdots\!50\)\( T^{9} + \)\(23\!\cdots\!10\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(65\!\cdots\!97\)\( T^{12} + \)\(17\!\cdots\!40\)\( p^{10} T^{13} + \)\(23\!\cdots\!10\)\( p^{20} T^{14} + \)\(21\!\cdots\!50\)\( p^{30} T^{15} + \)\(81\!\cdots\!46\)\( p^{40} T^{16} + \)\(15\!\cdots\!90\)\( p^{50} T^{17} + \)\(17\!\cdots\!78\)\( p^{60} T^{18} + \)\(16\!\cdots\!80\)\( p^{70} T^{19} + \)\(24\!\cdots\!53\)\( p^{80} T^{20} - \)\(81\!\cdots\!00\)\( p^{90} T^{21} + 208293091597001526 p^{100} T^{22} - 39022350 p^{110} T^{23} + p^{120} T^{24} \) | |
| 53 | \( 1 + 482396241 T - 342229819729012800 T^{2} - \)\(19\!\cdots\!63\)\( T^{3} + \)\(40\!\cdots\!62\)\( T^{4} + \)\(31\!\cdots\!89\)\( T^{5} - \)\(85\!\cdots\!98\)\( T^{6} - \)\(29\!\cdots\!01\)\( T^{7} - \)\(49\!\cdots\!70\)\( T^{8} + \)\(31\!\cdots\!15\)\( T^{9} - \)\(16\!\cdots\!36\)\( p T^{10} + \)\(24\!\cdots\!11\)\( T^{11} + \)\(50\!\cdots\!30\)\( T^{12} + \)\(24\!\cdots\!11\)\( p^{10} T^{13} - \)\(16\!\cdots\!36\)\( p^{21} T^{14} + \)\(31\!\cdots\!15\)\( p^{30} T^{15} - \)\(49\!\cdots\!70\)\( p^{40} T^{16} - \)\(29\!\cdots\!01\)\( p^{50} T^{17} - \)\(85\!\cdots\!98\)\( p^{60} T^{18} + \)\(31\!\cdots\!89\)\( p^{70} T^{19} + \)\(40\!\cdots\!62\)\( p^{80} T^{20} - \)\(19\!\cdots\!63\)\( p^{90} T^{21} - 342229819729012800 p^{100} T^{22} + 482396241 p^{110} T^{23} + p^{120} T^{24} \) | |
| 59 | \( 1 + 5699643 p T + 1989075301903149954 T^{2} + \)\(11\!\cdots\!33\)\( p T^{3} + \)\(19\!\cdots\!20\)\( T^{4} + \)\(17\!\cdots\!85\)\( T^{5} + \)\(12\!\cdots\!62\)\( T^{6} - \)\(27\!\cdots\!41\)\( T^{7} + \)\(71\!\cdots\!48\)\( T^{8} - \)\(25\!\cdots\!87\)\( T^{9} + \)\(44\!\cdots\!54\)\( T^{10} - \)\(14\!\cdots\!01\)\( T^{11} + \)\(26\!\cdots\!26\)\( T^{12} - \)\(14\!\cdots\!01\)\( p^{10} T^{13} + \)\(44\!\cdots\!54\)\( p^{20} T^{14} - \)\(25\!\cdots\!87\)\( p^{30} T^{15} + \)\(71\!\cdots\!48\)\( p^{40} T^{16} - \)\(27\!\cdots\!41\)\( p^{50} T^{17} + \)\(12\!\cdots\!62\)\( p^{60} T^{18} + \)\(17\!\cdots\!85\)\( p^{70} T^{19} + \)\(19\!\cdots\!20\)\( p^{80} T^{20} + \)\(11\!\cdots\!33\)\( p^{91} T^{21} + 1989075301903149954 p^{100} T^{22} + 5699643 p^{111} T^{23} + p^{120} T^{24} \) | |
| 61 | \( 1 - 586052496 T + 2454772981198080678 T^{2} - \)\(13\!\cdots\!76\)\( T^{3} + \)\(36\!\cdots\!09\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{5} + \)\(25\!\cdots\!78\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!50\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{9} - \)\(14\!\cdots\!66\)\( T^{10} + \)\(56\!\cdots\!68\)\( T^{11} - \)\(15\!\cdots\!11\)\( T^{12} + \)\(56\!\cdots\!68\)\( p^{10} T^{13} - \)\(14\!\cdots\!66\)\( p^{20} T^{14} - \)\(18\!\cdots\!84\)\( p^{30} T^{15} + \)\(48\!\cdots\!50\)\( p^{40} T^{16} - \)\(10\!\cdots\!00\)\( p^{50} T^{17} + \)\(25\!\cdots\!78\)\( p^{60} T^{18} - \)\(17\!\cdots\!52\)\( p^{70} T^{19} + \)\(36\!\cdots\!09\)\( p^{80} T^{20} - \)\(13\!\cdots\!76\)\( p^{90} T^{21} + 2454772981198080678 p^{100} T^{22} - 586052496 p^{110} T^{23} + p^{120} T^{24} \) | |
| 67 | \( 1 - 255759963 T - 5414166768037366380 T^{2} + \)\(87\!\cdots\!17\)\( T^{3} + \)\(13\!\cdots\!62\)\( T^{4} - \)\(38\!\cdots\!31\)\( T^{5} + \)\(28\!\cdots\!50\)\( T^{6} + \)\(79\!\cdots\!39\)\( T^{7} - \)\(69\!\cdots\!54\)\( T^{8} - \)\(78\!\cdots\!05\)\( T^{9} + \)\(14\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!75\)\( T^{11} - \)\(23\!\cdots\!82\)\( T^{12} + \)\(18\!\cdots\!75\)\( p^{10} T^{13} + \)\(14\!\cdots\!20\)\( p^{20} T^{14} - \)\(78\!\cdots\!05\)\( p^{30} T^{15} - \)\(69\!\cdots\!54\)\( p^{40} T^{16} + \)\(79\!\cdots\!39\)\( p^{50} T^{17} + \)\(28\!\cdots\!50\)\( p^{60} T^{18} - \)\(38\!\cdots\!31\)\( p^{70} T^{19} + \)\(13\!\cdots\!62\)\( p^{80} T^{20} + \)\(87\!\cdots\!17\)\( p^{90} T^{21} - 5414166768037366380 p^{100} T^{22} - 255759963 p^{110} T^{23} + p^{120} T^{24} \) | |
| 71 | \( ( 1 + 938097486 T + 11324820693073871130 T^{2} + \)\(72\!\cdots\!82\)\( T^{3} + \)\(64\!\cdots\!27\)\( T^{4} + \)\(32\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!92\)\( T^{6} + \)\(32\!\cdots\!36\)\( p^{10} T^{7} + \)\(64\!\cdots\!27\)\( p^{20} T^{8} + \)\(72\!\cdots\!82\)\( p^{30} T^{9} + 11324820693073871130 p^{40} T^{10} + 938097486 p^{50} T^{11} + p^{60} T^{12} )^{2} \) | |
| 73 | \( 1 + 4375723095 T + 29762072423030253606 T^{2} + \)\(10\!\cdots\!45\)\( T^{3} + \)\(42\!\cdots\!88\)\( T^{4} + \)\(11\!\cdots\!55\)\( T^{5} + \)\(38\!\cdots\!38\)\( T^{6} + \)\(90\!\cdots\!25\)\( T^{7} + \)\(25\!\cdots\!56\)\( T^{8} + \)\(51\!\cdots\!75\)\( T^{9} + \)\(12\!\cdots\!30\)\( T^{10} + \)\(24\!\cdots\!65\)\( T^{11} + \)\(58\!\cdots\!62\)\( T^{12} + \)\(24\!\cdots\!65\)\( p^{10} T^{13} + \)\(12\!\cdots\!30\)\( p^{20} T^{14} + \)\(51\!\cdots\!75\)\( p^{30} T^{15} + \)\(25\!\cdots\!56\)\( p^{40} T^{16} + \)\(90\!\cdots\!25\)\( p^{50} T^{17} + \)\(38\!\cdots\!38\)\( p^{60} T^{18} + \)\(11\!\cdots\!55\)\( p^{70} T^{19} + \)\(42\!\cdots\!88\)\( p^{80} T^{20} + \)\(10\!\cdots\!45\)\( p^{90} T^{21} + 29762072423030253606 p^{100} T^{22} + 4375723095 p^{110} T^{23} + p^{120} T^{24} \) | |
| 79 | \( 1 - 506220648 T - 42018009593120318523 T^{2} - \)\(66\!\cdots\!92\)\( T^{3} + \)\(95\!\cdots\!11\)\( T^{4} + \)\(49\!\cdots\!72\)\( T^{5} - \)\(15\!\cdots\!72\)\( T^{6} - \)\(85\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!25\)\( T^{8} + \)\(78\!\cdots\!76\)\( T^{9} - \)\(21\!\cdots\!17\)\( T^{10} - \)\(31\!\cdots\!48\)\( T^{11} + \)\(21\!\cdots\!26\)\( T^{12} - \)\(31\!\cdots\!48\)\( p^{10} T^{13} - \)\(21\!\cdots\!17\)\( p^{20} T^{14} + \)\(78\!\cdots\!76\)\( p^{30} T^{15} + \)\(19\!\cdots\!25\)\( p^{40} T^{16} - \)\(85\!\cdots\!40\)\( p^{50} T^{17} - \)\(15\!\cdots\!72\)\( p^{60} T^{18} + \)\(49\!\cdots\!72\)\( p^{70} T^{19} + \)\(95\!\cdots\!11\)\( p^{80} T^{20} - \)\(66\!\cdots\!92\)\( p^{90} T^{21} - 42018009593120318523 p^{100} T^{22} - 506220648 p^{110} T^{23} + p^{120} T^{24} \) | |
| 83 | \( 1 - 59606731294504195575 T^{2} + \)\(21\!\cdots\!23\)\( T^{4} - \)\(58\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!89\)\( T^{8} - \)\(25\!\cdots\!61\)\( T^{10} + \)\(42\!\cdots\!78\)\( T^{12} - \)\(25\!\cdots\!61\)\( p^{20} T^{14} + \)\(13\!\cdots\!89\)\( p^{40} T^{16} - \)\(58\!\cdots\!24\)\( p^{60} T^{18} + \)\(21\!\cdots\!23\)\( p^{80} T^{20} - 59606731294504195575 p^{100} T^{22} + p^{120} T^{24} \) | |
| 89 | \( 1 - 14058905598 T + \)\(13\!\cdots\!34\)\( T^{2} - \)\(11\!\cdots\!12\)\( p T^{3} + \)\(52\!\cdots\!05\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{5} + \)\(95\!\cdots\!22\)\( T^{6} - \)\(54\!\cdots\!06\)\( T^{7} + \)\(44\!\cdots\!58\)\( T^{8} - \)\(30\!\cdots\!42\)\( T^{9} + \)\(12\!\cdots\!74\)\( T^{10} - \)\(32\!\cdots\!76\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} - \)\(32\!\cdots\!76\)\( p^{10} T^{13} + \)\(12\!\cdots\!74\)\( p^{20} T^{14} - \)\(30\!\cdots\!42\)\( p^{30} T^{15} + \)\(44\!\cdots\!58\)\( p^{40} T^{16} - \)\(54\!\cdots\!06\)\( p^{50} T^{17} + \)\(95\!\cdots\!22\)\( p^{60} T^{18} - \)\(19\!\cdots\!00\)\( p^{70} T^{19} + \)\(52\!\cdots\!05\)\( p^{80} T^{20} - \)\(11\!\cdots\!12\)\( p^{91} T^{21} + \)\(13\!\cdots\!34\)\( p^{100} T^{22} - 14058905598 p^{110} T^{23} + p^{120} T^{24} \) | |
| 97 | \( 1 - \)\(23\!\cdots\!19\)\( T^{2} + \)\(25\!\cdots\!11\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{6} - \)\(22\!\cdots\!31\)\( T^{8} + \)\(11\!\cdots\!59\)\( T^{10} - \)\(11\!\cdots\!02\)\( T^{12} + \)\(11\!\cdots\!59\)\( p^{20} T^{14} - \)\(22\!\cdots\!31\)\( p^{40} T^{16} - \)\(13\!\cdots\!32\)\( p^{60} T^{18} + \)\(25\!\cdots\!11\)\( p^{80} T^{20} - \)\(23\!\cdots\!19\)\( p^{100} T^{22} + p^{120} T^{24} \) | |
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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
−2.43512170992280765629843338258, −2.38752033145956281166186387742, −2.28386053455327607628375129957, −2.22573328590877169173506513652, −1.87898776454059924395303312840, −1.87342319679476017175249021245, −1.86445166203513853777604476204, −1.77531490607900447871833723893, −1.76559682530155442324253305976, −1.68068198943030204765220513184, −1.66085742664707506902623845476, −1.64920991187259544455406123517, −1.44111750406875010667047023799, −1.23287476625091170608339381636, −0.923128561805254128905024044015, −0.922644606345402003965991443354, −0.896510950427605775573046645939, −0.77668463205322701135679062305, −0.74686000654882428850894565901, −0.64769552962943424841311852017, −0.35788879405741663691741066711, −0.22938272370163619353744944203, −0.06921755985511354501647632278, −0.05243682932178636936844221715, −0.00693256269768102549537883160, 0.00693256269768102549537883160, 0.05243682932178636936844221715, 0.06921755985511354501647632278, 0.22938272370163619353744944203, 0.35788879405741663691741066711, 0.64769552962943424841311852017, 0.74686000654882428850894565901, 0.77668463205322701135679062305, 0.896510950427605775573046645939, 0.922644606345402003965991443354, 0.923128561805254128905024044015, 1.23287476625091170608339381636, 1.44111750406875010667047023799, 1.64920991187259544455406123517, 1.66085742664707506902623845476, 1.68068198943030204765220513184, 1.76559682530155442324253305976, 1.77531490607900447871833723893, 1.86445166203513853777604476204, 1.87342319679476017175249021245, 1.87898776454059924395303312840, 2.22573328590877169173506513652, 2.28386053455327607628375129957, 2.38752033145956281166186387742, 2.43512170992280765629843338258
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.