Properties

Degree $28$
Conductor $3.169\times 10^{29}$
Sign $1$
Motivic weight $4$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 4·7-s + 2·9-s + 94·11-s + 2·13-s − 4·15-s − 4·17-s − 706·19-s − 8·21-s − 1.14e3·23-s + 2·25-s − 610·27-s − 862·29-s − 188·33-s + 8·35-s + 1.82e3·37-s − 4·39-s + 1.69e3·43-s + 4·45-s − 1.64e4·49-s + 8·51-s + 482·53-s + 188·55-s + 1.41e3·57-s − 2.78e3·59-s + 3.77e3·61-s + ⋯
L(s)  = 1  − 2/9·3-s + 2/25·5-s + 4/49·7-s + 2/81·9-s + 0.776·11-s + 0.0118·13-s − 0.0177·15-s − 0.0138·17-s − 1.95·19-s − 0.0181·21-s − 2.17·23-s + 0.00319·25-s − 0.836·27-s − 1.02·29-s − 0.172·33-s + 0.00653·35-s + 1.33·37-s − 0.00262·39-s + 0.916·43-s + 0.00197·45-s − 6.85·49-s + 0.00307·51-s + 0.171·53-s + 0.0621·55-s + 0.434·57-s − 0.800·59-s + 1.01·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{98}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{98}\right)^{s/2} \, \Gamma_{\C}(s+2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{98}\)
Sign: $1$
Motivic weight: \(4\)
Character: induced by $\chi_{128} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{98} ,\ ( \ : [2]^{14} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.000300430\)
\(L(\frac12)\) \(\approx\) \(0.000300430\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 2 T + 2 T^{2} + 610 T^{3} - 4613 T^{4} + 31732 T^{5} + 258740 T^{6} + 704956 p T^{7} + 7816417 p^{2} T^{8} + 4734970 p^{3} T^{9} + 12765802 p^{5} T^{10} + 91364362 p^{5} T^{11} + 11505403 p^{8} T^{12} + 372243416 p^{8} T^{13} + 3151621976 p^{8} T^{14} + 372243416 p^{12} T^{15} + 11505403 p^{16} T^{16} + 91364362 p^{17} T^{17} + 12765802 p^{21} T^{18} + 4734970 p^{23} T^{19} + 7816417 p^{26} T^{20} + 704956 p^{29} T^{21} + 258740 p^{32} T^{22} + 31732 p^{36} T^{23} - 4613 p^{40} T^{24} + 610 p^{44} T^{25} + 2 p^{48} T^{26} + 2 p^{52} T^{27} + p^{56} T^{28} \)
5 \( 1 - 2 T + 2 T^{2} - 3938 T^{3} - 18913 p T^{4} + 9371916 T^{5} - 2160156 p T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 112600777623819 p^{4} T^{12} - 53569344141254552 p^{2} T^{13} + 14585682673141755608 T^{14} - 53569344141254552 p^{6} T^{15} + 112600777623819 p^{12} T^{16} + 1149733079121218 p^{12} T^{17} + 18344031697054 p^{16} T^{18} - 563896366366 p^{20} T^{19} - 151886931511 p^{24} T^{20} + 6597908556 p^{28} T^{21} - 2160156 p^{33} T^{22} + 9371916 p^{36} T^{23} - 18913 p^{41} T^{24} - 3938 p^{44} T^{25} + 2 p^{48} T^{26} - 2 p^{52} T^{27} + p^{56} T^{28} \)
7 \( ( 1 - 2 T + 8235 T^{2} - 64404 T^{3} + 38860249 T^{4} - 447351454 T^{5} + 125587217723 T^{6} - 196872839848 p T^{7} + 125587217723 p^{4} T^{8} - 447351454 p^{8} T^{9} + 38860249 p^{12} T^{10} - 64404 p^{16} T^{11} + 8235 p^{20} T^{12} - 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
11 \( 1 - 94 T + 4418 T^{2} + 965570 T^{3} - 470088133 T^{4} + 283723804 p T^{5} + 2249641670708 T^{6} - 796542815073292 T^{7} + 97369777194709097 T^{8} + 5317394377195027646 T^{9} - \)\(63\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!86\)\( T^{11} - \)\(19\!\cdots\!01\)\( p^{2} T^{12} - \)\(20\!\cdots\!32\)\( p T^{13} + \)\(18\!\cdots\!92\)\( T^{14} - \)\(20\!\cdots\!32\)\( p^{5} T^{15} - \)\(19\!\cdots\!01\)\( p^{10} T^{16} + \)\(14\!\cdots\!86\)\( p^{12} T^{17} - \)\(63\!\cdots\!78\)\( p^{16} T^{18} + 5317394377195027646 p^{20} T^{19} + 97369777194709097 p^{24} T^{20} - 796542815073292 p^{28} T^{21} + 2249641670708 p^{32} T^{22} + 283723804 p^{37} T^{23} - 470088133 p^{40} T^{24} + 965570 p^{44} T^{25} + 4418 p^{48} T^{26} - 94 p^{52} T^{27} + p^{56} T^{28} \)
13 \( 1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 1832149307204 p T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(78\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} - \)\(22\!\cdots\!32\)\( p^{4} T^{15} + \)\(74\!\cdots\!35\)\( p^{8} T^{16} - \)\(14\!\cdots\!90\)\( p^{12} T^{17} + \)\(78\!\cdots\!46\)\( p^{16} T^{18} - \)\(10\!\cdots\!70\)\( p^{20} T^{19} + 1142495439970904649 p^{24} T^{20} - 16199073445624116 p^{28} T^{21} + 1832149307204 p^{33} T^{22} - 65707775476 p^{36} T^{23} + 1464853339 p^{40} T^{24} - 6883234 p^{44} T^{25} + 2 p^{48} T^{26} - 2 p^{52} T^{27} + p^{56} T^{28} \)
17 \( ( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 3318766585 p T^{4} - 174198961346 p T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} + 6454907058757691 p^{4} T^{8} - 174198961346 p^{9} T^{9} + 3318766585 p^{13} T^{10} - 12776716 p^{16} T^{11} + 333755 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
19 \( 1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} - \)\(71\!\cdots\!35\)\( T^{8} - \)\(22\!\cdots\!14\)\( T^{9} - \)\(57\!\cdots\!66\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(19\!\cdots\!61\)\( T^{12} + \)\(38\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} + \)\(38\!\cdots\!56\)\( p^{4} T^{15} - \)\(19\!\cdots\!61\)\( p^{8} T^{16} - \)\(18\!\cdots\!22\)\( p^{12} T^{17} - \)\(57\!\cdots\!66\)\( p^{16} T^{18} - \)\(22\!\cdots\!14\)\( p^{20} T^{19} - \)\(71\!\cdots\!35\)\( p^{24} T^{20} - 960761925731564364 p^{28} T^{21} - 431513784802892 p^{32} T^{22} + 3617133340788 p^{36} T^{23} + 32599242619 p^{40} T^{24} + 101381538 p^{44} T^{25} + 249218 p^{48} T^{26} + 706 p^{52} T^{27} + p^{56} T^{28} \)
23 \( ( 1 + 574 T + 1024043 T^{2} + 635922028 T^{3} + 551773439769 T^{4} + 314763003369506 T^{5} + 213700110666561659 T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + 213700110666561659 p^{4} T^{8} + 314763003369506 p^{8} T^{9} + 551773439769 p^{12} T^{10} + 635922028 p^{16} T^{11} + 1024043 p^{20} T^{12} + 574 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
29 \( 1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} + \)\(98\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!49\)\( T^{8} + \)\(31\!\cdots\!74\)\( T^{9} + \)\(39\!\cdots\!98\)\( T^{10} + \)\(15\!\cdots\!86\)\( p T^{11} + \)\(34\!\cdots\!87\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!96\)\( T^{14} + \)\(29\!\cdots\!64\)\( p^{4} T^{15} + \)\(34\!\cdots\!87\)\( p^{8} T^{16} + \)\(15\!\cdots\!86\)\( p^{13} T^{17} + \)\(39\!\cdots\!98\)\( p^{16} T^{18} + \)\(31\!\cdots\!74\)\( p^{20} T^{19} + \)\(75\!\cdots\!49\)\( p^{24} T^{20} + \)\(98\!\cdots\!28\)\( p^{28} T^{21} + 502604487474844916 p^{32} T^{22} + 691721668187596 p^{36} T^{23} + 1721779716827 p^{40} T^{24} + 1045006654 p^{44} T^{25} + 371522 p^{48} T^{26} + 862 p^{52} T^{27} + p^{56} T^{28} \)
31 \( 1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(57\!\cdots\!93\)\( p T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(57\!\cdots\!93\)\( p^{9} T^{16} - \)\(14\!\cdots\!50\)\( p^{16} T^{18} + \)\(10\!\cdots\!69\)\( p^{24} T^{20} - 57459081771770667372 p^{32} T^{22} + 24182883262811 p^{40} T^{24} - 6904334 p^{48} T^{26} + p^{56} T^{28} \)
37 \( 1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 102662887807662052 p T^{6} - \)\(11\!\cdots\!60\)\( T^{7} - \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!22\)\( T^{9} + \)\(49\!\cdots\!50\)\( p T^{10} + \)\(21\!\cdots\!18\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} + \)\(14\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!04\)\( T^{14} + \)\(14\!\cdots\!60\)\( p^{4} T^{15} - \)\(10\!\cdots\!97\)\( p^{8} T^{16} + \)\(21\!\cdots\!18\)\( p^{12} T^{17} + \)\(49\!\cdots\!50\)\( p^{17} T^{18} + \)\(16\!\cdots\!22\)\( p^{20} T^{19} - \)\(10\!\cdots\!63\)\( p^{24} T^{20} - \)\(11\!\cdots\!60\)\( p^{28} T^{21} + 102662887807662052 p^{33} T^{22} - 1716272691299380 p^{36} T^{23} + 7030206539163 p^{40} T^{24} - 4976934274 p^{44} T^{25} + 1667138 p^{48} T^{26} - 1826 p^{52} T^{27} + p^{56} T^{28} \)
41 \( 1 - 24523982 T^{2} + 302442312166171 T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(70\!\cdots\!70\)\( T^{10} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(84\!\cdots\!88\)\( T^{14} + \)\(27\!\cdots\!51\)\( p^{8} T^{16} - \)\(70\!\cdots\!70\)\( p^{16} T^{18} + \)\(14\!\cdots\!01\)\( p^{24} T^{20} - \)\(24\!\cdots\!76\)\( p^{32} T^{22} + 302442312166171 p^{40} T^{24} - 24523982 p^{48} T^{26} + p^{56} T^{28} \)
43 \( 1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} - \)\(38\!\cdots\!44\)\( T^{7} + \)\(23\!\cdots\!81\)\( T^{8} + \)\(78\!\cdots\!82\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} + \)\(10\!\cdots\!08\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} + \)\(10\!\cdots\!08\)\( p^{4} T^{15} - \)\(91\!\cdots\!45\)\( p^{8} T^{16} - \)\(18\!\cdots\!22\)\( p^{12} T^{17} + \)\(12\!\cdots\!62\)\( p^{16} T^{18} + \)\(78\!\cdots\!82\)\( p^{20} T^{19} + \)\(23\!\cdots\!81\)\( p^{24} T^{20} - \)\(38\!\cdots\!44\)\( p^{28} T^{21} + 60977294006539554100 p^{32} T^{22} - 13474016894363980 p^{36} T^{23} + 44454402050619 p^{40} T^{24} - 14278395262 p^{44} T^{25} + 1434818 p^{48} T^{26} - 1694 p^{52} T^{27} + p^{56} T^{28} \)
47 \( 1 - 51887758 T^{2} + 1309844227745755 T^{4} - \)\(45\!\cdots\!48\)\( p T^{6} + \)\(24\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(82\!\cdots\!04\)\( T^{14} + \)\(15\!\cdots\!39\)\( p^{8} T^{16} - \)\(21\!\cdots\!30\)\( p^{16} T^{18} + \)\(24\!\cdots\!09\)\( p^{24} T^{20} - \)\(45\!\cdots\!48\)\( p^{33} T^{22} + 1309844227745755 p^{40} T^{24} - 51887758 p^{48} T^{26} + p^{56} T^{28} \)
53 \( 1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!13\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(18\!\cdots\!24\)\( p^{4} T^{15} + \)\(28\!\cdots\!31\)\( p^{8} T^{16} - \)\(13\!\cdots\!10\)\( p^{12} T^{17} + \)\(12\!\cdots\!18\)\( p^{16} T^{18} + \)\(92\!\cdots\!94\)\( p^{20} T^{19} + \)\(35\!\cdots\!13\)\( p^{24} T^{20} - \)\(11\!\cdots\!76\)\( p^{28} T^{21} + 85293132817975457012 p^{32} T^{22} - 155411473123116980 p^{36} T^{23} + 43583027341595 p^{40} T^{24} + 5558326078 p^{44} T^{25} + 116162 p^{48} T^{26} - 482 p^{52} T^{27} + p^{56} T^{28} \)
59 \( 1 + 2786 T + 3880898 T^{2} + 95235375746 T^{3} + 282835430943931 T^{4} - 951817240129082700 T^{5} + \)\(78\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!24\)\( T^{7} - \)\(10\!\cdots\!59\)\( T^{8} - \)\(16\!\cdots\!78\)\( T^{9} + \)\(95\!\cdots\!50\)\( T^{10} - \)\(27\!\cdots\!94\)\( T^{11} + \)\(66\!\cdots\!23\)\( T^{12} + \)\(40\!\cdots\!76\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} + \)\(40\!\cdots\!76\)\( p^{4} T^{15} + \)\(66\!\cdots\!23\)\( p^{8} T^{16} - \)\(27\!\cdots\!94\)\( p^{12} T^{17} + \)\(95\!\cdots\!50\)\( p^{16} T^{18} - \)\(16\!\cdots\!78\)\( p^{20} T^{19} - \)\(10\!\cdots\!59\)\( p^{24} T^{20} - \)\(10\!\cdots\!24\)\( p^{28} T^{21} + \)\(78\!\cdots\!20\)\( p^{32} T^{22} - 951817240129082700 p^{36} T^{23} + 282835430943931 p^{40} T^{24} + 95235375746 p^{44} T^{25} + 3880898 p^{48} T^{26} + 2786 p^{52} T^{27} + p^{56} T^{28} \)
61 \( 1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(34\!\cdots\!59\)\( T^{8} + \)\(57\!\cdots\!82\)\( T^{9} - \)\(99\!\cdots\!50\)\( p T^{10} + \)\(13\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} - \)\(19\!\cdots\!60\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} - \)\(19\!\cdots\!60\)\( p^{4} T^{15} + \)\(15\!\cdots\!55\)\( p^{8} T^{16} + \)\(13\!\cdots\!70\)\( p^{12} T^{17} - \)\(99\!\cdots\!50\)\( p^{17} T^{18} + \)\(57\!\cdots\!82\)\( p^{20} T^{19} - \)\(34\!\cdots\!59\)\( p^{24} T^{20} + \)\(11\!\cdots\!80\)\( p^{28} T^{21} - \)\(12\!\cdots\!20\)\( p^{32} T^{22} + 322904339201283724 p^{36} T^{23} + 15885658886619 p^{40} T^{24} - 13584988130 p^{44} T^{25} + 7136642 p^{48} T^{26} - 3778 p^{52} T^{27} + p^{56} T^{28} \)
67 \( 1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} + \)\(99\!\cdots\!64\)\( T^{6} - \)\(93\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!21\)\( T^{8} + \)\(69\!\cdots\!70\)\( T^{9} - \)\(55\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!74\)\( T^{11} - \)\(39\!\cdots\!49\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(39\!\cdots\!76\)\( T^{14} + \)\(17\!\cdots\!60\)\( p^{4} T^{15} - \)\(39\!\cdots\!49\)\( p^{8} T^{16} + \)\(21\!\cdots\!74\)\( p^{12} T^{17} - \)\(55\!\cdots\!14\)\( p^{16} T^{18} + \)\(69\!\cdots\!70\)\( p^{20} T^{19} + \)\(28\!\cdots\!21\)\( p^{24} T^{20} - \)\(93\!\cdots\!52\)\( p^{28} T^{21} + \)\(99\!\cdots\!64\)\( p^{32} T^{22} + 648765759780793716 p^{36} T^{23} - 439076236005637 p^{40} T^{24} - 47670849246 p^{44} T^{25} + 31984002 p^{48} T^{26} - 7998 p^{52} T^{27} + p^{56} T^{28} \)
71 \( ( 1 + 9982 T + 145017323 T^{2} + 1165310044396 T^{3} + 10014081489420185 T^{4} + 63641985337531169890 T^{5} + \)\(40\!\cdots\!59\)\( T^{6} + \)\(20\!\cdots\!72\)\( T^{7} + \)\(40\!\cdots\!59\)\( p^{4} T^{8} + 63641985337531169890 p^{8} T^{9} + 10014081489420185 p^{12} T^{10} + 1165310044396 p^{16} T^{11} + 145017323 p^{20} T^{12} + 9982 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
73 \( 1 - 168573838 T^{2} + 13353714116727067 T^{4} - \)\(70\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!98\)\( T^{10} + \)\(39\!\cdots\!07\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(39\!\cdots\!07\)\( p^{8} T^{16} - \)\(11\!\cdots\!98\)\( p^{16} T^{18} + \)\(30\!\cdots\!57\)\( p^{24} T^{20} - \)\(70\!\cdots\!96\)\( p^{32} T^{22} + 13353714116727067 p^{40} T^{24} - 168573838 p^{48} T^{26} + p^{56} T^{28} \)
79 \( 1 - 364033678 T^{2} + 65454647116587227 T^{4} - \)\(76\!\cdots\!56\)\( T^{6} + \)\(65\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(99\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!67\)\( p^{8} T^{16} - \)\(43\!\cdots\!10\)\( p^{16} T^{18} + \)\(65\!\cdots\!81\)\( p^{24} T^{20} - \)\(76\!\cdots\!56\)\( p^{32} T^{22} + 65454647116587227 p^{40} T^{24} - 364033678 p^{48} T^{26} + p^{56} T^{28} \)
83 \( 1 + 17282 T + 149333762 T^{2} + 1088719641698 T^{3} + 16964332297412731 T^{4} + \)\(21\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} + \)\(11\!\cdots\!62\)\( T^{9} + \)\(90\!\cdots\!74\)\( T^{10} + \)\(61\!\cdots\!82\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} + \)\(35\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} + \)\(35\!\cdots\!48\)\( p^{4} T^{15} + \)\(44\!\cdots\!67\)\( p^{8} T^{16} + \)\(61\!\cdots\!82\)\( p^{12} T^{17} + \)\(90\!\cdots\!74\)\( p^{16} T^{18} + \)\(11\!\cdots\!62\)\( p^{20} T^{19} + \)\(11\!\cdots\!61\)\( p^{24} T^{20} + \)\(12\!\cdots\!12\)\( p^{28} T^{21} + \)\(17\!\cdots\!52\)\( p^{32} T^{22} + \)\(21\!\cdots\!96\)\( p^{36} T^{23} + 16964332297412731 p^{40} T^{24} + 1088719641698 p^{44} T^{25} + 149333762 p^{48} T^{26} + 17282 p^{52} T^{27} + p^{56} T^{28} \)
89 \( 1 - 548528910 T^{2} + 149200943223060123 T^{4} - \)\(26\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!49\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!11\)\( p^{8} T^{16} - \)\(37\!\cdots\!50\)\( p^{16} T^{18} + \)\(35\!\cdots\!49\)\( p^{24} T^{20} - \)\(26\!\cdots\!16\)\( p^{32} T^{22} + 149200943223060123 p^{40} T^{24} - 548528910 p^{48} T^{26} + p^{56} T^{28} \)
97 \( ( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{7} + \)\(94\!\cdots\!67\)\( p^{4} T^{8} + 73269666487293981214 p^{8} T^{9} + 75114732161345545 p^{12} T^{10} + 251760181236 p^{16} T^{11} + 387850619 p^{20} T^{12} + 2 p^{24} T^{13} + p^{28} T^{14} )^{2} \)
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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

−3.44136663753688172213832381879, −3.42798132716934368461864979620, −3.23216144555519374258734826568, −3.19522180578392609807596019307, −3.13612307607975077017458026032, −3.03872290986641721271293481373, −2.87593749472730154089441073339, −2.39281634571362825484142718452, −2.34458125383439833380869397453, −2.21701928078806766835142824230, −2.20873140985842180013465386818, −2.20385623016150239100261330432, −2.04063070418753170461591560135, −1.96843901793091553169532659105, −1.65936930420181572104160360815, −1.58898800433732612502773552420, −1.31475066414417438437216437719, −1.27753374680287605346283838831, −1.18465291604728923377790811228, −1.02369355814731790407857049875, −0.69591804884145730657163591976, −0.68472825190667696835378444408, −0.17491578382766491059102145480, −0.02770355095177397624396912803, −0.02097019397243991078038220246, 0.02097019397243991078038220246, 0.02770355095177397624396912803, 0.17491578382766491059102145480, 0.68472825190667696835378444408, 0.69591804884145730657163591976, 1.02369355814731790407857049875, 1.18465291604728923377790811228, 1.27753374680287605346283838831, 1.31475066414417438437216437719, 1.58898800433732612502773552420, 1.65936930420181572104160360815, 1.96843901793091553169532659105, 2.04063070418753170461591560135, 2.20385623016150239100261330432, 2.20873140985842180013465386818, 2.21701928078806766835142824230, 2.34458125383439833380869397453, 2.39281634571362825484142718452, 2.87593749472730154089441073339, 3.03872290986641721271293481373, 3.13612307607975077017458026032, 3.19522180578392609807596019307, 3.23216144555519374258734826568, 3.42798132716934368461864979620, 3.44136663753688172213832381879

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.