Dirichlet series
| L(s) = 1 | − 17·5-s + 144·7-s + 565·11-s − 842·13-s − 52·17-s + 107·19-s − 700·23-s + 7.07e3·25-s − 1.29e4·29-s − 3.55e3·31-s − 2.44e3·35-s − 3.45e3·37-s + 1.51e4·41-s + 3.82e4·43-s − 1.21e4·47-s − 9.59e3·49-s + 3.75e4·53-s − 9.60e3·55-s − 3.90e3·59-s + 1.32e4·61-s + 1.43e4·65-s − 2.88e3·67-s − 1.32e5·71-s + 1.27e5·73-s + 8.13e4·77-s + 1.00e5·79-s + 1.46e5·83-s + ⋯ |
| L(s) = 1 | − 0.304·5-s + 1.11·7-s + 1.40·11-s − 1.38·13-s − 0.0436·17-s + 0.0679·19-s − 0.275·23-s + 2.26·25-s − 2.86·29-s − 0.663·31-s − 0.337·35-s − 0.414·37-s + 1.40·41-s + 3.15·43-s − 0.801·47-s − 0.570·49-s + 1.83·53-s − 0.428·55-s − 0.146·59-s + 0.457·61-s + 0.420·65-s − 0.0786·67-s − 3.13·71-s + 2.80·73-s + 1.56·77-s + 1.80·79-s + 2.33·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.78203\times 10^{22}\) |
| Root analytic conductor: | \(8.99074\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(19.08673060\) |
| \(L(\frac12)\) | \(\approx\) | \(19.08673060\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 144 T + 619 p^{2} T^{2} + 61608 p T^{3} + 421934 p^{2} T^{4} + 184078896 p^{3} T^{5} - 85723343 p^{5} T^{6} + 184078896 p^{8} T^{7} + 421934 p^{12} T^{8} + 61608 p^{16} T^{9} + 619 p^{22} T^{10} - 144 p^{25} T^{11} + p^{30} T^{12} \) | |
| good | 5 | \( 1 + 17 T - 1358 p T^{2} + 50271 T^{3} + 4240964 p T^{4} - 96686903 p T^{5} - 5625198742 T^{6} + 24080544987 p T^{7} - 31500086768976 p T^{8} + 7299904534134781 T^{9} + 90708471768396202 p T^{10} - 18707288983607255533 T^{11} - \)\(11\!\cdots\!54\)\( T^{12} - 18707288983607255533 p^{5} T^{13} + 90708471768396202 p^{11} T^{14} + 7299904534134781 p^{15} T^{15} - 31500086768976 p^{21} T^{16} + 24080544987 p^{26} T^{17} - 5625198742 p^{30} T^{18} - 96686903 p^{36} T^{19} + 4240964 p^{41} T^{20} + 50271 p^{45} T^{21} - 1358 p^{51} T^{22} + 17 p^{55} T^{23} + p^{60} T^{24} \) |
| 11 | \( 1 - 565 T - 254434 T^{2} + 333519781 T^{3} - 48874532754 T^{4} - 57468434651945 T^{5} + 28962172090457780 T^{6} - 881668117061005049 T^{7} - \)\(35\!\cdots\!66\)\( T^{8} + \)\(16\!\cdots\!57\)\( T^{9} - \)\(23\!\cdots\!94\)\( p T^{10} - \)\(15\!\cdots\!01\)\( T^{11} + \)\(12\!\cdots\!74\)\( T^{12} - \)\(15\!\cdots\!01\)\( p^{5} T^{13} - \)\(23\!\cdots\!94\)\( p^{11} T^{14} + \)\(16\!\cdots\!57\)\( p^{15} T^{15} - \)\(35\!\cdots\!66\)\( p^{20} T^{16} - 881668117061005049 p^{25} T^{17} + 28962172090457780 p^{30} T^{18} - 57468434651945 p^{35} T^{19} - 48874532754 p^{40} T^{20} + 333519781 p^{45} T^{21} - 254434 p^{50} T^{22} - 565 p^{55} T^{23} + p^{60} T^{24} \) | |
| 13 | \( ( 1 + 421 T + 864117 T^{2} + 679339396 T^{3} + 587064869591 T^{4} + 338999931036859 T^{5} + 300006611173095558 T^{6} + 338999931036859 p^{5} T^{7} + 587064869591 p^{10} T^{8} + 679339396 p^{15} T^{9} + 864117 p^{20} T^{10} + 421 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 17 | \( 1 + 52 T - 231830 T^{2} - 1184981200 T^{3} - 4310387625627 T^{4} + 298101611607560 T^{5} + 1567398545591933086 T^{6} + \)\(32\!\cdots\!24\)\( T^{7} + \)\(95\!\cdots\!50\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{9} - \)\(19\!\cdots\!74\)\( T^{10} - \)\(96\!\cdots\!88\)\( T^{11} - \)\(18\!\cdots\!11\)\( T^{12} - \)\(96\!\cdots\!88\)\( p^{5} T^{13} - \)\(19\!\cdots\!74\)\( p^{10} T^{14} - \)\(14\!\cdots\!24\)\( p^{15} T^{15} + \)\(95\!\cdots\!50\)\( p^{20} T^{16} + \)\(32\!\cdots\!24\)\( p^{25} T^{17} + 1567398545591933086 p^{30} T^{18} + 298101611607560 p^{35} T^{19} - 4310387625627 p^{40} T^{20} - 1184981200 p^{45} T^{21} - 231830 p^{50} T^{22} + 52 p^{55} T^{23} + p^{60} T^{24} \) | |
| 19 | \( 1 - 107 T - 272608 T^{2} + 10349778137 T^{3} - 6282397697808 T^{4} - 271897263713985 p T^{5} + 72201391892575534732 T^{6} - \)\(42\!\cdots\!39\)\( T^{7} - \)\(31\!\cdots\!32\)\( T^{8} + \)\(36\!\cdots\!93\)\( T^{9} - \)\(17\!\cdots\!12\)\( T^{10} - \)\(99\!\cdots\!93\)\( p T^{11} + \)\(15\!\cdots\!46\)\( T^{12} - \)\(99\!\cdots\!93\)\( p^{6} T^{13} - \)\(17\!\cdots\!12\)\( p^{10} T^{14} + \)\(36\!\cdots\!93\)\( p^{15} T^{15} - \)\(31\!\cdots\!32\)\( p^{20} T^{16} - \)\(42\!\cdots\!39\)\( p^{25} T^{17} + 72201391892575534732 p^{30} T^{18} - 271897263713985 p^{36} T^{19} - 6282397697808 p^{40} T^{20} + 10349778137 p^{45} T^{21} - 272608 p^{50} T^{22} - 107 p^{55} T^{23} + p^{60} T^{24} \) | |
| 23 | \( 1 + 700 T - 2460570 T^{2} - 31304452880 T^{3} - 47302113837195 T^{4} + 52001349686798680 T^{5} + \)\(11\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} - \)\(18\!\cdots\!10\)\( T^{8} - \)\(20\!\cdots\!80\)\( T^{9} - \)\(22\!\cdots\!50\)\( T^{10} + \)\(17\!\cdots\!40\)\( p T^{11} + \)\(42\!\cdots\!49\)\( T^{12} + \)\(17\!\cdots\!40\)\( p^{6} T^{13} - \)\(22\!\cdots\!50\)\( p^{10} T^{14} - \)\(20\!\cdots\!80\)\( p^{15} T^{15} - \)\(18\!\cdots\!10\)\( p^{20} T^{16} + \)\(11\!\cdots\!40\)\( p^{25} T^{17} + \)\(11\!\cdots\!98\)\( p^{30} T^{18} + 52001349686798680 p^{35} T^{19} - 47302113837195 p^{40} T^{20} - 31304452880 p^{45} T^{21} - 2460570 p^{50} T^{22} + 700 p^{55} T^{23} + p^{60} T^{24} \) | |
| 29 | \( ( 1 + 6479 T + 92417977 T^{2} + 490193747888 T^{3} + 4233516758230391 T^{4} + 17793063264587278205 T^{5} + \)\(11\!\cdots\!18\)\( T^{6} + 17793063264587278205 p^{5} T^{7} + 4233516758230391 p^{10} T^{8} + 490193747888 p^{15} T^{9} + 92417977 p^{20} T^{10} + 6479 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 31 | \( 1 + 3552 T - 141423075 T^{2} - 318825721392 T^{3} + 11965899526668219 T^{4} + 16712985244897612152 T^{5} - \)\(70\!\cdots\!92\)\( T^{6} - \)\(55\!\cdots\!04\)\( T^{7} + \)\(32\!\cdots\!45\)\( T^{8} + \)\(11\!\cdots\!84\)\( T^{9} - \)\(12\!\cdots\!89\)\( T^{10} - \)\(12\!\cdots\!04\)\( T^{11} + \)\(37\!\cdots\!42\)\( T^{12} - \)\(12\!\cdots\!04\)\( p^{5} T^{13} - \)\(12\!\cdots\!89\)\( p^{10} T^{14} + \)\(11\!\cdots\!84\)\( p^{15} T^{15} + \)\(32\!\cdots\!45\)\( p^{20} T^{16} - \)\(55\!\cdots\!04\)\( p^{25} T^{17} - \)\(70\!\cdots\!92\)\( p^{30} T^{18} + 16712985244897612152 p^{35} T^{19} + 11965899526668219 p^{40} T^{20} - 318825721392 p^{45} T^{21} - 141423075 p^{50} T^{22} + 3552 p^{55} T^{23} + p^{60} T^{24} \) | |
| 37 | \( 1 + 3453 T - 180449216 T^{2} - 1856454401187 T^{3} + 14590326498589922 T^{4} + \)\(28\!\cdots\!17\)\( T^{5} + \)\(45\!\cdots\!54\)\( p T^{6} - \)\(24\!\cdots\!49\)\( T^{7} - \)\(13\!\cdots\!18\)\( T^{8} + \)\(12\!\cdots\!91\)\( T^{9} + \)\(16\!\cdots\!56\)\( T^{10} - \)\(29\!\cdots\!01\)\( T^{11} - \)\(12\!\cdots\!66\)\( T^{12} - \)\(29\!\cdots\!01\)\( p^{5} T^{13} + \)\(16\!\cdots\!56\)\( p^{10} T^{14} + \)\(12\!\cdots\!91\)\( p^{15} T^{15} - \)\(13\!\cdots\!18\)\( p^{20} T^{16} - \)\(24\!\cdots\!49\)\( p^{25} T^{17} + \)\(45\!\cdots\!54\)\( p^{31} T^{18} + \)\(28\!\cdots\!17\)\( p^{35} T^{19} + 14590326498589922 p^{40} T^{20} - 1856454401187 p^{45} T^{21} - 180449216 p^{50} T^{22} + 3453 p^{55} T^{23} + p^{60} T^{24} \) | |
| 41 | \( ( 1 - 7574 T + 389292450 T^{2} - 162879990054 T^{3} + 44420336464847855 T^{4} + \)\(46\!\cdots\!80\)\( T^{5} + \)\(29\!\cdots\!64\)\( T^{6} + \)\(46\!\cdots\!80\)\( p^{5} T^{7} + 44420336464847855 p^{10} T^{8} - 162879990054 p^{15} T^{9} + 389292450 p^{20} T^{10} - 7574 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 19139 T + 415294135 T^{2} - 6731246958234 T^{3} + 81470660647325393 T^{4} - \)\(10\!\cdots\!31\)\( T^{5} + \)\(12\!\cdots\!98\)\( T^{6} - \)\(10\!\cdots\!31\)\( p^{5} T^{7} + 81470660647325393 p^{10} T^{8} - 6731246958234 p^{15} T^{9} + 415294135 p^{20} T^{10} - 19139 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 47 | \( 1 + 12136 T - 945072014 T^{2} - 9870971533632 T^{3} + 502298469754267189 T^{4} + \)\(41\!\cdots\!36\)\( T^{5} - \)\(20\!\cdots\!18\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(68\!\cdots\!58\)\( T^{8} + \)\(23\!\cdots\!12\)\( T^{9} - \)\(19\!\cdots\!94\)\( T^{10} - \)\(21\!\cdots\!72\)\( T^{11} + \)\(48\!\cdots\!41\)\( T^{12} - \)\(21\!\cdots\!72\)\( p^{5} T^{13} - \)\(19\!\cdots\!94\)\( p^{10} T^{14} + \)\(23\!\cdots\!12\)\( p^{15} T^{15} + \)\(68\!\cdots\!58\)\( p^{20} T^{16} - \)\(11\!\cdots\!72\)\( p^{25} T^{17} - \)\(20\!\cdots\!18\)\( p^{30} T^{18} + \)\(41\!\cdots\!36\)\( p^{35} T^{19} + 502298469754267189 p^{40} T^{20} - 9870971533632 p^{45} T^{21} - 945072014 p^{50} T^{22} + 12136 p^{55} T^{23} + p^{60} T^{24} \) | |
| 53 | \( 1 - 37591 T + 596089732 T^{2} - 11856560312747 T^{3} + 295655153925746094 T^{4} - \)\(12\!\cdots\!55\)\( T^{5} - \)\(11\!\cdots\!18\)\( p T^{6} - \)\(23\!\cdots\!57\)\( T^{7} + \)\(18\!\cdots\!58\)\( T^{8} - \)\(32\!\cdots\!57\)\( T^{9} + \)\(43\!\cdots\!40\)\( T^{10} - \)\(13\!\cdots\!97\)\( T^{11} + \)\(14\!\cdots\!78\)\( T^{12} - \)\(13\!\cdots\!97\)\( p^{5} T^{13} + \)\(43\!\cdots\!40\)\( p^{10} T^{14} - \)\(32\!\cdots\!57\)\( p^{15} T^{15} + \)\(18\!\cdots\!58\)\( p^{20} T^{16} - \)\(23\!\cdots\!57\)\( p^{25} T^{17} - \)\(11\!\cdots\!18\)\( p^{31} T^{18} - \)\(12\!\cdots\!55\)\( p^{35} T^{19} + 295655153925746094 p^{40} T^{20} - 11856560312747 p^{45} T^{21} + 596089732 p^{50} T^{22} - 37591 p^{55} T^{23} + p^{60} T^{24} \) | |
| 59 | \( 1 + 3905 T - 633718828 T^{2} - 21269904534695 T^{3} - 951848664888011484 T^{4} + \)\(11\!\cdots\!85\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} + \)\(94\!\cdots\!45\)\( T^{7} + \)\(65\!\cdots\!88\)\( T^{8} + \)\(74\!\cdots\!65\)\( T^{9} + \)\(92\!\cdots\!72\)\( T^{10} - \)\(95\!\cdots\!35\)\( T^{11} - \)\(52\!\cdots\!70\)\( T^{12} - \)\(95\!\cdots\!35\)\( p^{5} T^{13} + \)\(92\!\cdots\!72\)\( p^{10} T^{14} + \)\(74\!\cdots\!65\)\( p^{15} T^{15} + \)\(65\!\cdots\!88\)\( p^{20} T^{16} + \)\(94\!\cdots\!45\)\( p^{25} T^{17} + \)\(25\!\cdots\!08\)\( p^{30} T^{18} + \)\(11\!\cdots\!85\)\( p^{35} T^{19} - 951848664888011484 p^{40} T^{20} - 21269904534695 p^{45} T^{21} - 633718828 p^{50} T^{22} + 3905 p^{55} T^{23} + p^{60} T^{24} \) | |
| 61 | \( 1 - 13296 T - 1443995666 T^{2} + 37490357243008 T^{3} - 83203054698398755 T^{4} - \)\(16\!\cdots\!32\)\( T^{5} + \)\(99\!\cdots\!94\)\( T^{6} - \)\(25\!\cdots\!76\)\( T^{7} + \)\(43\!\cdots\!50\)\( T^{8} + \)\(29\!\cdots\!76\)\( T^{9} - \)\(50\!\cdots\!50\)\( T^{10} - \)\(82\!\cdots\!12\)\( T^{11} + \)\(34\!\cdots\!97\)\( T^{12} - \)\(82\!\cdots\!12\)\( p^{5} T^{13} - \)\(50\!\cdots\!50\)\( p^{10} T^{14} + \)\(29\!\cdots\!76\)\( p^{15} T^{15} + \)\(43\!\cdots\!50\)\( p^{20} T^{16} - \)\(25\!\cdots\!76\)\( p^{25} T^{17} + \)\(99\!\cdots\!94\)\( p^{30} T^{18} - \)\(16\!\cdots\!32\)\( p^{35} T^{19} - 83203054698398755 p^{40} T^{20} + 37490357243008 p^{45} T^{21} - 1443995666 p^{50} T^{22} - 13296 p^{55} T^{23} + p^{60} T^{24} \) | |
| 67 | \( 1 + 2889 T - 4941391098 T^{2} + 1121463078263 T^{3} + 12852543540454223910 T^{4} - \)\(34\!\cdots\!15\)\( T^{5} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{7} + \)\(23\!\cdots\!98\)\( T^{8} - \)\(21\!\cdots\!97\)\( T^{9} - \)\(18\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!49\)\( T^{11} + \)\(16\!\cdots\!54\)\( T^{12} + \)\(13\!\cdots\!49\)\( p^{5} T^{13} - \)\(18\!\cdots\!38\)\( p^{10} T^{14} - \)\(21\!\cdots\!97\)\( p^{15} T^{15} + \)\(23\!\cdots\!98\)\( p^{20} T^{16} + \)\(12\!\cdots\!61\)\( p^{25} T^{17} - \)\(21\!\cdots\!40\)\( p^{30} T^{18} - \)\(34\!\cdots\!15\)\( p^{35} T^{19} + 12852543540454223910 p^{40} T^{20} + 1121463078263 p^{45} T^{21} - 4941391098 p^{50} T^{22} + 2889 p^{55} T^{23} + p^{60} T^{24} \) | |
| 71 | \( ( 1 + 66496 T + 7366102350 T^{2} + 284630547907200 T^{3} + 20461246318357100735 T^{4} + \)\(59\!\cdots\!60\)\( T^{5} + \)\(40\!\cdots\!52\)\( T^{6} + \)\(59\!\cdots\!60\)\( p^{5} T^{7} + 20461246318357100735 p^{10} T^{8} + 284630547907200 p^{15} T^{9} + 7366102350 p^{20} T^{10} + 66496 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 73 | \( 1 - 127931 T + 2477894930 T^{2} + 201492479366923 T^{3} + 4854458306416229000 T^{4} - \)\(11\!\cdots\!23\)\( T^{5} + \)\(31\!\cdots\!66\)\( p T^{6} + \)\(12\!\cdots\!27\)\( T^{7} - \)\(51\!\cdots\!60\)\( T^{8} - \)\(14\!\cdots\!07\)\( T^{9} + \)\(15\!\cdots\!70\)\( T^{10} + \)\(28\!\cdots\!35\)\( T^{11} - \)\(53\!\cdots\!66\)\( T^{12} + \)\(28\!\cdots\!35\)\( p^{5} T^{13} + \)\(15\!\cdots\!70\)\( p^{10} T^{14} - \)\(14\!\cdots\!07\)\( p^{15} T^{15} - \)\(51\!\cdots\!60\)\( p^{20} T^{16} + \)\(12\!\cdots\!27\)\( p^{25} T^{17} + \)\(31\!\cdots\!66\)\( p^{31} T^{18} - \)\(11\!\cdots\!23\)\( p^{35} T^{19} + 4854458306416229000 p^{40} T^{20} + 201492479366923 p^{45} T^{21} + 2477894930 p^{50} T^{22} - 127931 p^{55} T^{23} + p^{60} T^{24} \) | |
| 79 | \( 1 - 100226 T - 10432607209 T^{2} + 846350121285610 T^{3} + \)\(11\!\cdots\!67\)\( T^{4} - \)\(57\!\cdots\!20\)\( T^{5} - \)\(74\!\cdots\!64\)\( T^{6} + \)\(22\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!29\)\( T^{8} - \)\(61\!\cdots\!06\)\( T^{9} - \)\(16\!\cdots\!67\)\( T^{10} + \)\(75\!\cdots\!74\)\( T^{11} + \)\(55\!\cdots\!86\)\( T^{12} + \)\(75\!\cdots\!74\)\( p^{5} T^{13} - \)\(16\!\cdots\!67\)\( p^{10} T^{14} - \)\(61\!\cdots\!06\)\( p^{15} T^{15} + \)\(39\!\cdots\!29\)\( p^{20} T^{16} + \)\(22\!\cdots\!40\)\( p^{25} T^{17} - \)\(74\!\cdots\!64\)\( p^{30} T^{18} - \)\(57\!\cdots\!20\)\( p^{35} T^{19} + \)\(11\!\cdots\!67\)\( p^{40} T^{20} + 846350121285610 p^{45} T^{21} - 10432607209 p^{50} T^{22} - 100226 p^{55} T^{23} + p^{60} T^{24} \) | |
| 83 | \( ( 1 - 881 p T + 10109038407 T^{2} - 719825774455058 T^{3} + 64104680711454127393 T^{4} - \)\(37\!\cdots\!51\)\( T^{5} + \)\(31\!\cdots\!90\)\( T^{6} - \)\(37\!\cdots\!51\)\( p^{5} T^{7} + 64104680711454127393 p^{10} T^{8} - 719825774455058 p^{15} T^{9} + 10109038407 p^{20} T^{10} - 881 p^{26} T^{11} + p^{30} T^{12} )^{2} \) | |
| 89 | \( 1 - 3418 T - 19217448798 T^{2} + 920528069590624 T^{3} + \)\(17\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{5} - \)\(79\!\cdots\!94\)\( T^{6} + \)\(68\!\cdots\!22\)\( T^{7} + \)\(33\!\cdots\!26\)\( T^{8} - \)\(63\!\cdots\!86\)\( T^{9} - \)\(38\!\cdots\!30\)\( T^{10} - \)\(34\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!05\)\( T^{12} - \)\(34\!\cdots\!40\)\( p^{5} T^{13} - \)\(38\!\cdots\!30\)\( p^{10} T^{14} - \)\(63\!\cdots\!86\)\( p^{15} T^{15} + \)\(33\!\cdots\!26\)\( p^{20} T^{16} + \)\(68\!\cdots\!22\)\( p^{25} T^{17} - \)\(79\!\cdots\!94\)\( p^{30} T^{18} - \)\(13\!\cdots\!72\)\( p^{35} T^{19} + \)\(17\!\cdots\!37\)\( p^{40} T^{20} + 920528069590624 p^{45} T^{21} - 19217448798 p^{50} T^{22} - 3418 p^{55} T^{23} + p^{60} T^{24} \) | |
| 97 | \( ( 1 - 14467 T + 34905497299 T^{2} - 1034744309037444 T^{3} + \)\(60\!\cdots\!69\)\( T^{4} - \)\(17\!\cdots\!93\)\( T^{5} + \)\(65\!\cdots\!18\)\( T^{6} - \)\(17\!\cdots\!93\)\( p^{5} T^{7} + \)\(60\!\cdots\!69\)\( p^{10} T^{8} - 1034744309037444 p^{15} T^{9} + 34905497299 p^{20} T^{10} - 14467 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
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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
−2.64204658050692548622044890169, −2.56192433127385251952001186189, −2.51732239339440352899742447499, −2.51650682311416130497233847604, −2.29110087150045973963414612228, −2.23145298883179219725099166231, −1.99503589441068060779081005527, −1.93469444286782508878502773927, −1.89090809707812635337101190609, −1.88628250544223239224869984351, −1.86654185446000584184382066713, −1.39284434380049771715282418517, −1.32861954915664182070670364665, −1.32626751496188393095757993002, −1.30485781671765514169824585581, −1.21765727511211785011781422320, −1.12341130693584118715651631525, −0.817802810347576260277062112321, −0.66366157667339002389797331938, −0.62523953650982622887798843153, −0.60869763004220589226522629184, −0.53259619081293397636040758668, −0.27640961999883345849239432237, −0.18748281029743206991681363928, −0.15473570349812960796611918654, 0.15473570349812960796611918654, 0.18748281029743206991681363928, 0.27640961999883345849239432237, 0.53259619081293397636040758668, 0.60869763004220589226522629184, 0.62523953650982622887798843153, 0.66366157667339002389797331938, 0.817802810347576260277062112321, 1.12341130693584118715651631525, 1.21765727511211785011781422320, 1.30485781671765514169824585581, 1.32626751496188393095757993002, 1.32861954915664182070670364665, 1.39284434380049771715282418517, 1.86654185446000584184382066713, 1.88628250544223239224869984351, 1.89090809707812635337101190609, 1.93469444286782508878502773927, 1.99503589441068060779081005527, 2.23145298883179219725099166231, 2.29110087150045973963414612228, 2.51650682311416130497233847604, 2.51732239339440352899742447499, 2.56192433127385251952001186189, 2.64204658050692548622044890169
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.