Properties

Label 28-12e14-1.1-c14e14-0-0
Degree $28$
Conductor $1.284\times 10^{15}$
Sign $1$
Analytic cond. $2.70738\times 10^{16}$
Root an. cond. $3.86257$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 182·2-s + 2.12e4·4-s − 1.61e4·5-s + 3.30e6·8-s − 1.11e7·9-s − 2.93e6·10-s + 1.09e8·13-s + 4.92e8·16-s − 2.91e8·17-s − 2.03e9·18-s − 3.42e8·20-s − 3.63e10·25-s + 2.00e10·26-s − 1.21e10·29-s + 7.00e10·32-s − 5.30e10·34-s − 2.36e11·36-s + 1.19e11·37-s − 5.32e10·40-s + 1.89e11·41-s + 1.79e11·45-s + 4.36e12·49-s − 6.61e12·50-s + 2.33e12·52-s + 1.25e12·53-s − 2.20e12·58-s − 7.88e12·61-s + ⋯
L(s)  = 1  + 1.42·2-s + 1.29·4-s − 0.206·5-s + 1.57·8-s − 7/3·9-s − 0.293·10-s + 1.75·13-s + 1.83·16-s − 0.710·17-s − 3.31·18-s − 0.267·20-s − 5.95·25-s + 2.49·26-s − 0.702·29-s + 2.03·32-s − 1.01·34-s − 3.02·36-s + 1.25·37-s − 0.325·40-s + 0.972·41-s + 0.481·45-s + 6.43·49-s − 8.46·50-s + 2.26·52-s + 1.06·53-s − 0.999·58-s − 2.50·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(15-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14}\right)^{s/2} \, \Gamma_{\C}(s+7)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{28} \cdot 3^{14}\)
Sign: $1$
Analytic conductor: \(2.70738\times 10^{16}\)
Root analytic conductor: \(3.86257\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{28} \cdot 3^{14} ,\ ( \ : [7]^{14} ),\ 1 )\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(21.85890449\)
\(L(\frac12)\) \(\approx\) \(21.85890449\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 91 p T + 2977 p^{2} T^{2} - 50271 p^{5} T^{3} + 579609 p^{8} T^{4} - 1527447 p^{13} T^{5} - 1134035 p^{18} T^{6} + 14229209 p^{25} T^{7} - 1134035 p^{32} T^{8} - 1527447 p^{41} T^{9} + 579609 p^{50} T^{10} - 50271 p^{61} T^{11} + 2977 p^{72} T^{12} - 91 p^{85} T^{13} + p^{98} T^{14} \)
3 \( ( 1 + p^{13} T^{2} )^{7} \)
good5 \( ( 1 + 8062 T + 3653617559 p T^{2} + 87286525877628 p T^{3} + 7282873467703616769 p^{2} T^{4} + \)\(18\!\cdots\!98\)\( p^{2} T^{5} + \)\(42\!\cdots\!79\)\( p^{5} T^{6} + \)\(20\!\cdots\!08\)\( p^{6} T^{7} + \)\(42\!\cdots\!79\)\( p^{19} T^{8} + \)\(18\!\cdots\!98\)\( p^{30} T^{9} + 7282873467703616769 p^{44} T^{10} + 87286525877628 p^{57} T^{11} + 3653617559 p^{71} T^{12} + 8062 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
7 \( 1 - 4362986924318 T^{2} + \)\(13\!\cdots\!21\)\( p T^{4} - \)\(27\!\cdots\!96\)\( p^{2} T^{6} + \)\(44\!\cdots\!75\)\( p^{3} T^{8} - \)\(60\!\cdots\!94\)\( p^{4} T^{10} + \)\(70\!\cdots\!21\)\( p^{5} T^{12} - \)\(72\!\cdots\!84\)\( p^{6} T^{14} + \)\(70\!\cdots\!21\)\( p^{33} T^{16} - \)\(60\!\cdots\!94\)\( p^{60} T^{18} + \)\(44\!\cdots\!75\)\( p^{87} T^{20} - \)\(27\!\cdots\!96\)\( p^{114} T^{22} + \)\(13\!\cdots\!21\)\( p^{141} T^{24} - 4362986924318 p^{168} T^{26} + p^{196} T^{28} \)
11 \( 1 - 3233446705923374 T^{2} + \)\(49\!\cdots\!43\)\( T^{4} - \)\(39\!\cdots\!00\)\( p^{2} T^{6} + \)\(23\!\cdots\!73\)\( p^{4} T^{8} - \)\(10\!\cdots\!26\)\( p^{6} T^{10} + \)\(39\!\cdots\!71\)\( p^{8} T^{12} - \)\(13\!\cdots\!64\)\( p^{10} T^{14} + \)\(39\!\cdots\!71\)\( p^{36} T^{16} - \)\(10\!\cdots\!26\)\( p^{62} T^{18} + \)\(23\!\cdots\!73\)\( p^{88} T^{20} - \)\(39\!\cdots\!00\)\( p^{114} T^{22} + \)\(49\!\cdots\!43\)\( p^{140} T^{24} - 3233446705923374 p^{168} T^{26} + p^{196} T^{28} \)
13 \( ( 1 - 54967070 T + 1316516371729231 p T^{2} - \)\(46\!\cdots\!96\)\( p^{2} T^{3} + \)\(48\!\cdots\!57\)\( p^{4} T^{4} - \)\(19\!\cdots\!30\)\( p^{4} T^{5} + \)\(19\!\cdots\!83\)\( p^{5} T^{6} - \)\(55\!\cdots\!56\)\( p^{6} T^{7} + \)\(19\!\cdots\!83\)\( p^{19} T^{8} - \)\(19\!\cdots\!30\)\( p^{32} T^{9} + \)\(48\!\cdots\!57\)\( p^{46} T^{10} - \)\(46\!\cdots\!96\)\( p^{58} T^{11} + 1316516371729231 p^{71} T^{12} - 54967070 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
17 \( ( 1 + 145741690 T + 554939623742304859 T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!85\)\( T^{4} + \)\(42\!\cdots\!90\)\( T^{5} + \)\(38\!\cdots\!23\)\( T^{6} + \)\(88\!\cdots\!40\)\( T^{7} + \)\(38\!\cdots\!23\)\( p^{14} T^{8} + \)\(42\!\cdots\!90\)\( p^{28} T^{9} + \)\(17\!\cdots\!85\)\( p^{42} T^{10} + \)\(12\!\cdots\!00\)\( p^{56} T^{11} + 554939623742304859 p^{70} T^{12} + 145741690 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
19 \( 1 - 169658841265658618 p T^{2} + \)\(60\!\cdots\!67\)\( T^{4} - \)\(91\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!65\)\( T^{8} - \)\(12\!\cdots\!86\)\( T^{10} + \)\(12\!\cdots\!07\)\( T^{12} - \)\(10\!\cdots\!84\)\( T^{14} + \)\(12\!\cdots\!07\)\( p^{28} T^{16} - \)\(12\!\cdots\!86\)\( p^{56} T^{18} + \)\(11\!\cdots\!65\)\( p^{84} T^{20} - \)\(91\!\cdots\!36\)\( p^{112} T^{22} + \)\(60\!\cdots\!67\)\( p^{140} T^{24} - 169658841265658618 p^{169} T^{26} + p^{196} T^{28} \)
23 \( 1 - 76574512257177783086 T^{2} + \)\(29\!\cdots\!19\)\( T^{4} - \)\(78\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!17\)\( T^{8} - \)\(24\!\cdots\!06\)\( T^{10} + \)\(34\!\cdots\!23\)\( T^{12} - \)\(41\!\cdots\!44\)\( T^{14} + \)\(34\!\cdots\!23\)\( p^{28} T^{16} - \)\(24\!\cdots\!06\)\( p^{56} T^{18} + \)\(15\!\cdots\!17\)\( p^{84} T^{20} - \)\(78\!\cdots\!40\)\( p^{112} T^{22} + \)\(29\!\cdots\!19\)\( p^{140} T^{24} - 76574512257177783086 p^{168} T^{26} + p^{196} T^{28} \)
29 \( ( 1 + 6063102406 T + \)\(11\!\cdots\!27\)\( T^{2} + \)\(73\!\cdots\!48\)\( T^{3} + \)\(62\!\cdots\!73\)\( T^{4} + \)\(45\!\cdots\!58\)\( T^{5} + \)\(24\!\cdots\!47\)\( T^{6} + \)\(16\!\cdots\!28\)\( T^{7} + \)\(24\!\cdots\!47\)\( p^{14} T^{8} + \)\(45\!\cdots\!58\)\( p^{28} T^{9} + \)\(62\!\cdots\!73\)\( p^{42} T^{10} + \)\(73\!\cdots\!48\)\( p^{56} T^{11} + \)\(11\!\cdots\!27\)\( p^{70} T^{12} + 6063102406 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
31 \( 1 - \)\(36\!\cdots\!62\)\( T^{2} + \)\(71\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!77\)\( T^{8} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!75\)\( T^{12} - \)\(10\!\cdots\!16\)\( T^{14} + \)\(12\!\cdots\!75\)\( p^{28} T^{16} - \)\(14\!\cdots\!42\)\( p^{56} T^{18} + \)\(13\!\cdots\!77\)\( p^{84} T^{20} - \)\(10\!\cdots\!64\)\( p^{112} T^{22} + \)\(71\!\cdots\!39\)\( p^{140} T^{24} - \)\(36\!\cdots\!62\)\( p^{168} T^{26} + p^{196} T^{28} \)
37 \( ( 1 - 59682850790 T + \)\(24\!\cdots\!43\)\( T^{2} - \)\(14\!\cdots\!48\)\( T^{3} + \)\(83\!\cdots\!61\)\( p T^{4} + \)\(63\!\cdots\!74\)\( T^{5} + \)\(32\!\cdots\!15\)\( T^{6} + \)\(97\!\cdots\!04\)\( T^{7} + \)\(32\!\cdots\!15\)\( p^{14} T^{8} + \)\(63\!\cdots\!74\)\( p^{28} T^{9} + \)\(83\!\cdots\!61\)\( p^{43} T^{10} - \)\(14\!\cdots\!48\)\( p^{56} T^{11} + \)\(24\!\cdots\!43\)\( p^{70} T^{12} - 59682850790 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
41 \( ( 1 - 94659446966 T + \)\(67\!\cdots\!27\)\( T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!85\)\( T^{4} + \)\(62\!\cdots\!46\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(20\!\cdots\!03\)\( p^{14} T^{8} + \)\(62\!\cdots\!46\)\( p^{28} T^{9} + \)\(14\!\cdots\!85\)\( p^{42} T^{10} + \)\(11\!\cdots\!64\)\( p^{56} T^{11} + \)\(67\!\cdots\!27\)\( p^{70} T^{12} - 94659446966 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
43 \( 1 - \)\(51\!\cdots\!74\)\( T^{2} + \)\(14\!\cdots\!55\)\( T^{4} - \)\(27\!\cdots\!28\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} - \)\(48\!\cdots\!38\)\( T^{10} + \)\(47\!\cdots\!07\)\( T^{12} - \)\(38\!\cdots\!64\)\( T^{14} + \)\(47\!\cdots\!07\)\( p^{28} T^{16} - \)\(48\!\cdots\!38\)\( p^{56} T^{18} + \)\(41\!\cdots\!21\)\( p^{84} T^{20} - \)\(27\!\cdots\!28\)\( p^{112} T^{22} + \)\(14\!\cdots\!55\)\( p^{140} T^{24} - \)\(51\!\cdots\!74\)\( p^{168} T^{26} + p^{196} T^{28} \)
47 \( 1 - \)\(14\!\cdots\!94\)\( T^{2} + \)\(10\!\cdots\!63\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!13\)\( T^{8} - \)\(80\!\cdots\!26\)\( T^{10} + \)\(24\!\cdots\!59\)\( T^{12} - \)\(30\!\cdots\!96\)\( p^{2} T^{14} + \)\(24\!\cdots\!59\)\( p^{28} T^{16} - \)\(80\!\cdots\!26\)\( p^{56} T^{18} + \)\(22\!\cdots\!13\)\( p^{84} T^{20} - \)\(55\!\cdots\!92\)\( p^{112} T^{22} + \)\(10\!\cdots\!63\)\( p^{140} T^{24} - \)\(14\!\cdots\!94\)\( p^{168} T^{26} + p^{196} T^{28} \)
53 \( ( 1 - 625695945482 T + \)\(46\!\cdots\!35\)\( T^{2} - \)\(23\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!53\)\( T^{4} - \)\(55\!\cdots\!34\)\( T^{5} + \)\(21\!\cdots\!99\)\( T^{6} - \)\(94\!\cdots\!28\)\( T^{7} + \)\(21\!\cdots\!99\)\( p^{14} T^{8} - \)\(55\!\cdots\!34\)\( p^{28} T^{9} + \)\(11\!\cdots\!53\)\( p^{42} T^{10} - \)\(23\!\cdots\!32\)\( p^{56} T^{11} + \)\(46\!\cdots\!35\)\( p^{70} T^{12} - 625695945482 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
59 \( 1 - \)\(31\!\cdots\!22\)\( T^{2} + \)\(54\!\cdots\!87\)\( T^{4} - \)\(61\!\cdots\!56\)\( T^{6} + \)\(50\!\cdots\!25\)\( T^{8} - \)\(32\!\cdots\!66\)\( T^{10} + \)\(17\!\cdots\!67\)\( T^{12} - \)\(97\!\cdots\!64\)\( T^{14} + \)\(17\!\cdots\!67\)\( p^{28} T^{16} - \)\(32\!\cdots\!66\)\( p^{56} T^{18} + \)\(50\!\cdots\!25\)\( p^{84} T^{20} - \)\(61\!\cdots\!56\)\( p^{112} T^{22} + \)\(54\!\cdots\!87\)\( p^{140} T^{24} - \)\(31\!\cdots\!22\)\( p^{168} T^{26} + p^{196} T^{28} \)
61 \( ( 1 + 3941220838330 T + \)\(48\!\cdots\!35\)\( T^{2} + \)\(14\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!65\)\( T^{4} + \)\(25\!\cdots\!66\)\( T^{5} + \)\(15\!\cdots\!63\)\( T^{6} + \)\(30\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!63\)\( p^{14} T^{8} + \)\(25\!\cdots\!66\)\( p^{28} T^{9} + \)\(10\!\cdots\!65\)\( p^{42} T^{10} + \)\(14\!\cdots\!28\)\( p^{56} T^{11} + \)\(48\!\cdots\!35\)\( p^{70} T^{12} + 3941220838330 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
67 \( 1 - \)\(24\!\cdots\!78\)\( T^{2} + \)\(29\!\cdots\!47\)\( T^{4} - \)\(24\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!45\)\( T^{8} - \)\(81\!\cdots\!74\)\( T^{10} + \)\(36\!\cdots\!47\)\( T^{12} - \)\(14\!\cdots\!16\)\( T^{14} + \)\(36\!\cdots\!47\)\( p^{28} T^{16} - \)\(81\!\cdots\!74\)\( p^{56} T^{18} + \)\(15\!\cdots\!45\)\( p^{84} T^{20} - \)\(24\!\cdots\!44\)\( p^{112} T^{22} + \)\(29\!\cdots\!47\)\( p^{140} T^{24} - \)\(24\!\cdots\!78\)\( p^{168} T^{26} + p^{196} T^{28} \)
71 \( 1 - \)\(36\!\cdots\!62\)\( T^{2} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(34\!\cdots\!80\)\( T^{6} - \)\(25\!\cdots\!79\)\( T^{8} + \)\(16\!\cdots\!58\)\( T^{10} - \)\(39\!\cdots\!53\)\( T^{12} - \)\(45\!\cdots\!20\)\( T^{14} - \)\(39\!\cdots\!53\)\( p^{28} T^{16} + \)\(16\!\cdots\!58\)\( p^{56} T^{18} - \)\(25\!\cdots\!79\)\( p^{84} T^{20} - \)\(34\!\cdots\!80\)\( p^{112} T^{22} + \)\(53\!\cdots\!95\)\( p^{140} T^{24} - \)\(36\!\cdots\!62\)\( p^{168} T^{26} + p^{196} T^{28} \)
73 \( ( 1 - 19592531125214 T + \)\(59\!\cdots\!91\)\( T^{2} - \)\(79\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} - \)\(12\!\cdots\!62\)\( T^{5} + \)\(14\!\cdots\!31\)\( T^{6} - \)\(14\!\cdots\!36\)\( T^{7} + \)\(14\!\cdots\!31\)\( p^{14} T^{8} - \)\(12\!\cdots\!62\)\( p^{28} T^{9} + \)\(12\!\cdots\!81\)\( p^{42} T^{10} - \)\(79\!\cdots\!48\)\( p^{56} T^{11} + \)\(59\!\cdots\!91\)\( p^{70} T^{12} - 19592531125214 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
79 \( 1 - \)\(26\!\cdots\!30\)\( T^{2} + \)\(36\!\cdots\!11\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(24\!\cdots\!45\)\( T^{8} - \)\(13\!\cdots\!30\)\( T^{10} + \)\(65\!\cdots\!47\)\( T^{12} - \)\(26\!\cdots\!00\)\( T^{14} + \)\(65\!\cdots\!47\)\( p^{28} T^{16} - \)\(13\!\cdots\!30\)\( p^{56} T^{18} + \)\(24\!\cdots\!45\)\( p^{84} T^{20} - \)\(33\!\cdots\!60\)\( p^{112} T^{22} + \)\(36\!\cdots\!11\)\( p^{140} T^{24} - \)\(26\!\cdots\!30\)\( p^{168} T^{26} + p^{196} T^{28} \)
83 \( 1 - \)\(81\!\cdots\!42\)\( T^{2} + \)\(31\!\cdots\!47\)\( T^{4} - \)\(78\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!05\)\( T^{8} - \)\(18\!\cdots\!78\)\( T^{10} + \)\(19\!\cdots\!91\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!91\)\( p^{28} T^{16} - \)\(18\!\cdots\!78\)\( p^{56} T^{18} + \)\(13\!\cdots\!05\)\( p^{84} T^{20} - \)\(78\!\cdots\!92\)\( p^{112} T^{22} + \)\(31\!\cdots\!47\)\( p^{140} T^{24} - \)\(81\!\cdots\!42\)\( p^{168} T^{26} + p^{196} T^{28} \)
89 \( ( 1 - 111860666992286 T + \)\(16\!\cdots\!95\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!97\)\( T^{4} - \)\(58\!\cdots\!78\)\( T^{5} + \)\(34\!\cdots\!11\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(34\!\cdots\!11\)\( p^{14} T^{8} - \)\(58\!\cdots\!78\)\( p^{28} T^{9} + \)\(10\!\cdots\!97\)\( p^{42} T^{10} - \)\(12\!\cdots\!60\)\( p^{56} T^{11} + \)\(16\!\cdots\!95\)\( p^{70} T^{12} - 111860666992286 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
97 \( ( 1 - 141451140180878 T + \)\(26\!\cdots\!11\)\( T^{2} - \)\(31\!\cdots\!48\)\( T^{3} + \)\(40\!\cdots\!73\)\( T^{4} - \)\(38\!\cdots\!62\)\( T^{5} + \)\(38\!\cdots\!99\)\( T^{6} - \)\(30\!\cdots\!12\)\( T^{7} + \)\(38\!\cdots\!99\)\( p^{14} T^{8} - \)\(38\!\cdots\!62\)\( p^{28} T^{9} + \)\(40\!\cdots\!73\)\( p^{42} T^{10} - \)\(31\!\cdots\!48\)\( p^{56} T^{11} + \)\(26\!\cdots\!11\)\( p^{70} T^{12} - 141451140180878 p^{84} T^{13} + p^{98} T^{14} )^{2} \)
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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

−3.94455565056224553200235615904, −3.89992251230670121743464875561, −3.64486287061581745947291715005, −3.46843298766986930891094852079, −3.36123023395864350153872346175, −3.35912583917873494346698816902, −3.16147880790139276784026160665, −2.77247471790145324950175232888, −2.71263290411705165689678988560, −2.46832089886442467792302793943, −2.36388626580380895000155630817, −2.28527134796179441735373767320, −2.11428846272502706070912592516, −1.97702384705913224038773911152, −1.83372699957173972261085122724, −1.71572991322867274284609255670, −1.57914296167866866850417456272, −1.36810460191124394755642699685, −0.828582563106109730248591414011, −0.78570172223900011479665779760, −0.70180695971945335582345905618, −0.61872308499268369511177399400, −0.53561527303860276226257884081, −0.47303149145376726215686738507, −0.10729446479312400363926981633, 0.10729446479312400363926981633, 0.47303149145376726215686738507, 0.53561527303860276226257884081, 0.61872308499268369511177399400, 0.70180695971945335582345905618, 0.78570172223900011479665779760, 0.828582563106109730248591414011, 1.36810460191124394755642699685, 1.57914296167866866850417456272, 1.71572991322867274284609255670, 1.83372699957173972261085122724, 1.97702384705913224038773911152, 2.11428846272502706070912592516, 2.28527134796179441735373767320, 2.36388626580380895000155630817, 2.46832089886442467792302793943, 2.71263290411705165689678988560, 2.77247471790145324950175232888, 3.16147880790139276784026160665, 3.35912583917873494346698816902, 3.36123023395864350153872346175, 3.46843298766986930891094852079, 3.64486287061581745947291715005, 3.89992251230670121743464875561, 3.94455565056224553200235615904

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

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