Properties

Degree $28$
Conductor $268435456$
Sign $1$
Motivic weight $32$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37e4·2-s − 1.17e9·4-s + 1.38e11·5-s − 3.12e13·8-s + 7.37e15·9-s − 3.28e15·10-s − 1.64e18·13-s − 5.45e18·16-s + 3.15e19·17-s − 1.75e20·18-s − 1.62e20·20-s − 1.48e23·25-s + 3.90e22·26-s − 3.83e23·29-s − 3.75e23·32-s − 7.50e23·34-s − 8.69e24·36-s − 1.07e25·37-s − 4.32e24·40-s + 3.63e25·41-s + 1.01e27·45-s + 5.31e27·49-s + 3.52e27·50-s + 1.93e27·52-s + 1.84e28·53-s + 9.11e27·58-s + 1.06e29·61-s + ⋯
L(s)  = 1  − 0.362·2-s − 0.274·4-s + 0.905·5-s − 0.111·8-s + 3.98·9-s − 0.328·10-s − 2.46·13-s − 0.295·16-s + 0.648·17-s − 1.44·18-s − 0.248·20-s − 6.37·25-s + 0.894·26-s − 1.53·29-s − 0.310·32-s − 0.235·34-s − 1.09·36-s − 0.870·37-s − 0.100·40-s + 0.569·41-s + 3.60·45-s + 4.81·49-s + 2.31·50-s + 0.676·52-s + 4.75·53-s + 0.555·58-s + 2.89·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(33-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+16)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{28}\)
Sign: $1$
Motivic weight: \(32\)
Character: induced by $\chi_{4} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{28} ,\ ( \ : [16]^{14} ),\ 1 )\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(7.381037169\)
\(L(\frac12)\) \(\approx\) \(7.381037169\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5945 p^{2} T + 27252631 p^{6} T^{2} + 24605160195 p^{12} T^{3} + 10160468307837 p^{20} T^{4} + 867575578811145 p^{30} T^{5} + 14974227775100648 p^{42} T^{6} - 38743735189032144 p^{56} T^{7} + 14974227775100648 p^{74} T^{8} + 867575578811145 p^{94} T^{9} + 10160468307837 p^{116} T^{10} + 24605160195 p^{140} T^{11} + 27252631 p^{166} T^{12} + 5945 p^{194} T^{13} + p^{224} T^{14} \)
good3 \( 1 - 2459795451851354 p T^{2} + \)\(13\!\cdots\!05\)\( p^{5} T^{4} - \)\(56\!\cdots\!44\)\( p^{11} T^{6} + \)\(72\!\cdots\!89\)\( p^{20} T^{8} - \)\(27\!\cdots\!70\)\( p^{30} T^{10} + \)\(35\!\cdots\!05\)\( p^{43} T^{12} - \)\(14\!\cdots\!20\)\( p^{57} T^{14} + \)\(35\!\cdots\!05\)\( p^{107} T^{16} - \)\(27\!\cdots\!70\)\( p^{158} T^{18} + \)\(72\!\cdots\!89\)\( p^{212} T^{20} - \)\(56\!\cdots\!44\)\( p^{267} T^{22} + \)\(13\!\cdots\!05\)\( p^{325} T^{24} - 2459795451851354 p^{385} T^{26} + p^{448} T^{28} \)
5 \( ( 1 - 13812149174 p T + \)\(65\!\cdots\!83\)\( p^{3} T^{2} - \)\(19\!\cdots\!32\)\( p^{5} T^{3} + \)\(91\!\cdots\!77\)\( p^{8} T^{4} - \)\(21\!\cdots\!54\)\( p^{13} T^{5} + \)\(14\!\cdots\!59\)\( p^{17} T^{6} - \)\(32\!\cdots\!24\)\( p^{22} T^{7} + \)\(14\!\cdots\!59\)\( p^{49} T^{8} - \)\(21\!\cdots\!54\)\( p^{77} T^{9} + \)\(91\!\cdots\!77\)\( p^{104} T^{10} - \)\(19\!\cdots\!32\)\( p^{133} T^{11} + \)\(65\!\cdots\!83\)\( p^{163} T^{12} - 13812149174 p^{193} T^{13} + p^{224} T^{14} )^{2} \)
7 \( 1 - \)\(75\!\cdots\!66\)\( p T^{2} + \)\(32\!\cdots\!15\)\( p^{2} T^{4} - \)\(10\!\cdots\!56\)\( p^{3} T^{6} + \)\(79\!\cdots\!23\)\( p^{7} T^{8} - \)\(72\!\cdots\!10\)\( p^{12} T^{10} + \)\(82\!\cdots\!75\)\( p^{18} T^{12} - \)\(16\!\cdots\!60\)\( p^{26} T^{14} + \)\(82\!\cdots\!75\)\( p^{82} T^{16} - \)\(72\!\cdots\!10\)\( p^{140} T^{18} + \)\(79\!\cdots\!23\)\( p^{199} T^{20} - \)\(10\!\cdots\!56\)\( p^{259} T^{22} + \)\(32\!\cdots\!15\)\( p^{322} T^{24} - \)\(75\!\cdots\!66\)\( p^{385} T^{26} + p^{448} T^{28} \)
11 \( 1 - \)\(12\!\cdots\!54\)\( p T^{2} + \)\(78\!\cdots\!71\)\( p^{2} T^{4} - \)\(35\!\cdots\!84\)\( p^{3} T^{6} + \)\(12\!\cdots\!41\)\( p^{4} T^{8} - \)\(36\!\cdots\!02\)\( p^{5} T^{10} + \)\(68\!\cdots\!93\)\( p^{9} T^{12} - \)\(10\!\cdots\!72\)\( p^{13} T^{14} + \)\(68\!\cdots\!93\)\( p^{73} T^{16} - \)\(36\!\cdots\!02\)\( p^{133} T^{18} + \)\(12\!\cdots\!41\)\( p^{196} T^{20} - \)\(35\!\cdots\!84\)\( p^{259} T^{22} + \)\(78\!\cdots\!71\)\( p^{322} T^{24} - \)\(12\!\cdots\!54\)\( p^{385} T^{26} + p^{448} T^{28} \)
13 \( ( 1 + 820209234838929010 T + \)\(18\!\cdots\!03\)\( p T^{2} + \)\(10\!\cdots\!00\)\( p^{2} T^{3} + \)\(12\!\cdots\!81\)\( p^{3} T^{4} + \)\(43\!\cdots\!10\)\( p^{5} T^{5} + \)\(30\!\cdots\!79\)\( p^{7} T^{6} + \)\(86\!\cdots\!60\)\( p^{9} T^{7} + \)\(30\!\cdots\!79\)\( p^{39} T^{8} + \)\(43\!\cdots\!10\)\( p^{69} T^{9} + \)\(12\!\cdots\!81\)\( p^{99} T^{10} + \)\(10\!\cdots\!00\)\( p^{130} T^{11} + \)\(18\!\cdots\!03\)\( p^{161} T^{12} + 820209234838929010 p^{192} T^{13} + p^{224} T^{14} )^{2} \)
17 \( ( 1 - 15785483952898628110 T + \)\(81\!\cdots\!39\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!21\)\( p T^{4} - \)\(18\!\cdots\!50\)\( p^{2} T^{5} + \)\(26\!\cdots\!31\)\( p^{3} T^{6} - \)\(17\!\cdots\!20\)\( p^{4} T^{7} + \)\(26\!\cdots\!31\)\( p^{35} T^{8} - \)\(18\!\cdots\!50\)\( p^{66} T^{9} + \)\(23\!\cdots\!21\)\( p^{97} T^{10} - \)\(11\!\cdots\!00\)\( p^{128} T^{11} + \)\(81\!\cdots\!39\)\( p^{160} T^{12} - 15785483952898628110 p^{192} T^{13} + p^{224} T^{14} )^{2} \)
19 \( 1 - \)\(56\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!91\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!81\)\( p^{2} T^{8} - \)\(38\!\cdots\!62\)\( p^{4} T^{10} + \)\(10\!\cdots\!63\)\( p^{6} T^{12} - \)\(26\!\cdots\!52\)\( p^{8} T^{14} + \)\(10\!\cdots\!63\)\( p^{70} T^{16} - \)\(38\!\cdots\!62\)\( p^{132} T^{18} + \)\(11\!\cdots\!81\)\( p^{194} T^{20} - \)\(29\!\cdots\!44\)\( p^{256} T^{22} + \)\(15\!\cdots\!91\)\( p^{320} T^{24} - \)\(56\!\cdots\!14\)\( p^{384} T^{26} + p^{448} T^{28} \)
23 \( 1 - \)\(39\!\cdots\!22\)\( T^{2} + \)\(75\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!12\)\( p^{2} T^{6} + \)\(27\!\cdots\!69\)\( p^{4} T^{8} - \)\(34\!\cdots\!10\)\( p^{6} T^{10} + \)\(33\!\cdots\!95\)\( p^{8} T^{12} - \)\(26\!\cdots\!80\)\( p^{10} T^{14} + \)\(33\!\cdots\!95\)\( p^{72} T^{16} - \)\(34\!\cdots\!10\)\( p^{134} T^{18} + \)\(27\!\cdots\!69\)\( p^{196} T^{20} - \)\(17\!\cdots\!12\)\( p^{258} T^{22} + \)\(75\!\cdots\!95\)\( p^{320} T^{24} - \)\(39\!\cdots\!22\)\( p^{384} T^{26} + p^{448} T^{28} \)
29 \( ( 1 + \)\(66\!\cdots\!02\)\( p T + \)\(26\!\cdots\!75\)\( p^{2} T^{2} + \)\(95\!\cdots\!88\)\( p^{3} T^{3} + \)\(26\!\cdots\!69\)\( p^{4} T^{4} + \)\(57\!\cdots\!30\)\( p^{5} T^{5} + \)\(14\!\cdots\!55\)\( p^{6} T^{6} - \)\(45\!\cdots\!40\)\( p^{7} T^{7} + \)\(14\!\cdots\!55\)\( p^{38} T^{8} + \)\(57\!\cdots\!30\)\( p^{69} T^{9} + \)\(26\!\cdots\!69\)\( p^{100} T^{10} + \)\(95\!\cdots\!88\)\( p^{131} T^{11} + \)\(26\!\cdots\!75\)\( p^{162} T^{12} + \)\(66\!\cdots\!02\)\( p^{193} T^{13} + p^{224} T^{14} )^{2} \)
31 \( 1 - \)\(83\!\cdots\!54\)\( p^{3} T^{2} + \)\(37\!\cdots\!71\)\( p^{4} T^{4} - \)\(37\!\cdots\!24\)\( p^{6} T^{6} + \)\(30\!\cdots\!01\)\( p^{8} T^{8} - \)\(19\!\cdots\!02\)\( p^{10} T^{10} + \)\(11\!\cdots\!23\)\( p^{12} T^{12} - \)\(66\!\cdots\!92\)\( p^{14} T^{14} + \)\(11\!\cdots\!23\)\( p^{76} T^{16} - \)\(19\!\cdots\!02\)\( p^{138} T^{18} + \)\(30\!\cdots\!01\)\( p^{200} T^{20} - \)\(37\!\cdots\!24\)\( p^{262} T^{22} + \)\(37\!\cdots\!71\)\( p^{324} T^{24} - \)\(83\!\cdots\!54\)\( p^{387} T^{26} + p^{448} T^{28} \)
37 \( ( 1 + \)\(53\!\cdots\!70\)\( T + \)\(85\!\cdots\!99\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!97\)\( T^{4} + \)\(13\!\cdots\!30\)\( T^{5} + \)\(78\!\cdots\!63\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(78\!\cdots\!63\)\( p^{32} T^{8} + \)\(13\!\cdots\!30\)\( p^{64} T^{9} + \)\(33\!\cdots\!97\)\( p^{96} T^{10} + \)\(40\!\cdots\!20\)\( p^{128} T^{11} + \)\(85\!\cdots\!99\)\( p^{160} T^{12} + \)\(53\!\cdots\!70\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
41 \( ( 1 - \)\(18\!\cdots\!74\)\( T + \)\(16\!\cdots\!11\)\( T^{2} - \)\(37\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(68\!\cdots\!23\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(68\!\cdots\!23\)\( p^{32} T^{8} - \)\(25\!\cdots\!02\)\( p^{64} T^{9} + \)\(12\!\cdots\!61\)\( p^{96} T^{10} - \)\(37\!\cdots\!44\)\( p^{128} T^{11} + \)\(16\!\cdots\!11\)\( p^{160} T^{12} - \)\(18\!\cdots\!74\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
43 \( 1 - \)\(10\!\cdots\!42\)\( T^{2} + \)\(66\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(10\!\cdots\!29\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{10} + \)\(71\!\cdots\!15\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(71\!\cdots\!15\)\( p^{64} T^{16} - \)\(29\!\cdots\!50\)\( p^{128} T^{18} + \)\(10\!\cdots\!29\)\( p^{192} T^{20} - \)\(29\!\cdots\!68\)\( p^{256} T^{22} + \)\(66\!\cdots\!55\)\( p^{320} T^{24} - \)\(10\!\cdots\!42\)\( p^{384} T^{26} + p^{448} T^{28} \)
47 \( 1 - \)\(28\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!89\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(47\!\cdots\!35\)\( T^{12} - \)\(35\!\cdots\!80\)\( p T^{14} + \)\(47\!\cdots\!35\)\( p^{64} T^{16} - \)\(11\!\cdots\!30\)\( p^{128} T^{18} + \)\(21\!\cdots\!89\)\( p^{192} T^{20} - \)\(33\!\cdots\!08\)\( p^{256} T^{22} + \)\(37\!\cdots\!15\)\( p^{320} T^{24} - \)\(28\!\cdots\!22\)\( p^{384} T^{26} + p^{448} T^{28} \)
53 \( ( 1 - \)\(92\!\cdots\!90\)\( T + \)\(83\!\cdots\!59\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!57\)\( T^{4} - \)\(11\!\cdots\!10\)\( T^{5} + \)\(54\!\cdots\!63\)\( T^{6} - \)\(20\!\cdots\!60\)\( T^{7} + \)\(54\!\cdots\!63\)\( p^{32} T^{8} - \)\(11\!\cdots\!10\)\( p^{64} T^{9} + \)\(26\!\cdots\!57\)\( p^{96} T^{10} - \)\(46\!\cdots\!20\)\( p^{128} T^{11} + \)\(83\!\cdots\!59\)\( p^{160} T^{12} - \)\(92\!\cdots\!90\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
59 \( 1 - \)\(29\!\cdots\!14\)\( T^{2} + \)\(31\!\cdots\!71\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} - \)\(51\!\cdots\!79\)\( T^{8} + \)\(42\!\cdots\!98\)\( T^{10} + \)\(82\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{14} + \)\(82\!\cdots\!03\)\( p^{64} T^{16} + \)\(42\!\cdots\!98\)\( p^{128} T^{18} - \)\(51\!\cdots\!79\)\( p^{192} T^{20} - \)\(10\!\cdots\!04\)\( p^{256} T^{22} + \)\(31\!\cdots\!71\)\( p^{320} T^{24} - \)\(29\!\cdots\!14\)\( p^{384} T^{26} + p^{448} T^{28} \)
61 \( ( 1 - \)\(53\!\cdots\!54\)\( T + \)\(85\!\cdots\!91\)\( p T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} - \)\(31\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!83\)\( T^{6} - \)\(55\!\cdots\!52\)\( T^{7} + \)\(22\!\cdots\!83\)\( p^{32} T^{8} - \)\(31\!\cdots\!02\)\( p^{64} T^{9} + \)\(11\!\cdots\!41\)\( p^{96} T^{10} - \)\(16\!\cdots\!84\)\( p^{128} T^{11} + \)\(85\!\cdots\!91\)\( p^{161} T^{12} - \)\(53\!\cdots\!54\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
67 \( 1 - \)\(17\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!89\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{10} + \)\(62\!\cdots\!15\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(62\!\cdots\!15\)\( p^{64} T^{16} - \)\(18\!\cdots\!10\)\( p^{128} T^{18} + \)\(48\!\cdots\!89\)\( p^{192} T^{20} - \)\(10\!\cdots\!68\)\( p^{256} T^{22} + \)\(16\!\cdots\!55\)\( p^{320} T^{24} - \)\(17\!\cdots\!62\)\( p^{384} T^{26} + p^{448} T^{28} \)
71 \( 1 - \)\(96\!\cdots\!14\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{6} + \)\(55\!\cdots\!61\)\( T^{8} - \)\(13\!\cdots\!02\)\( T^{10} + \)\(27\!\cdots\!03\)\( T^{12} - \)\(49\!\cdots\!72\)\( T^{14} + \)\(27\!\cdots\!03\)\( p^{64} T^{16} - \)\(13\!\cdots\!02\)\( p^{128} T^{18} + \)\(55\!\cdots\!61\)\( p^{192} T^{20} - \)\(19\!\cdots\!84\)\( p^{256} T^{22} + \)\(51\!\cdots\!11\)\( p^{320} T^{24} - \)\(96\!\cdots\!14\)\( p^{384} T^{26} + p^{448} T^{28} \)
73 \( ( 1 - \)\(71\!\cdots\!30\)\( T + \)\(19\!\cdots\!99\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!17\)\( T^{4} - \)\(89\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!03\)\( p^{32} T^{8} - \)\(89\!\cdots\!30\)\( p^{64} T^{9} + \)\(17\!\cdots\!17\)\( p^{96} T^{10} - \)\(11\!\cdots\!20\)\( p^{128} T^{11} + \)\(19\!\cdots\!99\)\( p^{160} T^{12} - \)\(71\!\cdots\!30\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
79 \( 1 - \)\(32\!\cdots\!34\)\( T^{2} + \)\(79\!\cdots\!69\)\( p T^{4} - \)\(84\!\cdots\!04\)\( T^{6} + \)\(88\!\cdots\!61\)\( T^{8} - \)\(73\!\cdots\!02\)\( T^{10} + \)\(50\!\cdots\!43\)\( T^{12} - \)\(29\!\cdots\!52\)\( T^{14} + \)\(50\!\cdots\!43\)\( p^{64} T^{16} - \)\(73\!\cdots\!02\)\( p^{128} T^{18} + \)\(88\!\cdots\!61\)\( p^{192} T^{20} - \)\(84\!\cdots\!04\)\( p^{256} T^{22} + \)\(79\!\cdots\!69\)\( p^{321} T^{24} - \)\(32\!\cdots\!34\)\( p^{384} T^{26} + p^{448} T^{28} \)
83 \( 1 - \)\(22\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(97\!\cdots\!69\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{10} + \)\(14\!\cdots\!95\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!95\)\( p^{64} T^{16} - \)\(41\!\cdots\!90\)\( p^{128} T^{18} + \)\(97\!\cdots\!69\)\( p^{192} T^{20} - \)\(17\!\cdots\!08\)\( p^{256} T^{22} + \)\(24\!\cdots\!95\)\( p^{320} T^{24} - \)\(22\!\cdots\!02\)\( p^{384} T^{26} + p^{448} T^{28} \)
89 \( ( 1 + \)\(92\!\cdots\!38\)\( T + \)\(73\!\cdots\!55\)\( T^{2} + \)\(12\!\cdots\!92\)\( T^{3} + \)\(36\!\cdots\!89\)\( T^{4} + \)\(55\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!15\)\( p^{32} T^{8} + \)\(55\!\cdots\!50\)\( p^{64} T^{9} + \)\(36\!\cdots\!89\)\( p^{96} T^{10} + \)\(12\!\cdots\!92\)\( p^{128} T^{11} + \)\(73\!\cdots\!55\)\( p^{160} T^{12} + \)\(92\!\cdots\!38\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
97 \( ( 1 + \)\(10\!\cdots\!10\)\( T + \)\(16\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(42\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!63\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!63\)\( p^{32} T^{8} + \)\(42\!\cdots\!30\)\( p^{64} T^{9} + \)\(10\!\cdots\!57\)\( p^{96} T^{10} + \)\(11\!\cdots\!20\)\( p^{128} T^{11} + \)\(16\!\cdots\!39\)\( p^{160} T^{12} + \)\(10\!\cdots\!10\)\( p^{192} T^{13} + p^{224} T^{14} )^{2} \)
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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

−3.55178030838392514253090057749, −3.39142768925149755064810627838, −3.17310513816787989147301863144, −2.75936959394890153380692633726, −2.49280493029866675264390903314, −2.49002079785047663088228635367, −2.48502706778180948986390503951, −2.43750937467631057657562525146, −2.19917284530657578357194319583, −2.13755417494897341293527274274, −1.86054716661326145017518195316, −1.84309669527337363891195207279, −1.82302346163115416272555654209, −1.73766525367221020102943367279, −1.47761061364304727613233228297, −1.32574118559947280449017266610, −1.28125965015893501421822210043, −0.991190937236505938020415310219, −0.76509847696168848584030493330, −0.65727447862748033665491306779, −0.65475533794856822958827552831, −0.65440036514173282534173907012, −0.31902615374232653937731624382, −0.27698848101185851346731347330, −0.098759474866183931409339251022, 0.098759474866183931409339251022, 0.27698848101185851346731347330, 0.31902615374232653937731624382, 0.65440036514173282534173907012, 0.65475533794856822958827552831, 0.65727447862748033665491306779, 0.76509847696168848584030493330, 0.991190937236505938020415310219, 1.28125965015893501421822210043, 1.32574118559947280449017266610, 1.47761061364304727613233228297, 1.73766525367221020102943367279, 1.82302346163115416272555654209, 1.84309669527337363891195207279, 1.86054716661326145017518195316, 2.13755417494897341293527274274, 2.19917284530657578357194319583, 2.43750937467631057657562525146, 2.48502706778180948986390503951, 2.49002079785047663088228635367, 2.49280493029866675264390903314, 2.75936959394890153380692633726, 3.17310513816787989147301863144, 3.39142768925149755064810627838, 3.55178030838392514253090057749

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

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