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Properties

Label 12-304e6-1.1-c9e6-0-1
Degree 1212
Conductor 7.893×10147.893\times 10^{14}
Sign 11
Analytic cond. 1.47321×10131.47321\times 10^{13}
Root an. cond. 12.512812.5128
Motivic weight 99
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 155·3-s − 3.61e3·5-s − 4.08e3·7-s − 3.82e4·9-s + 6.93e4·11-s − 1.91e5·13-s − 5.59e5·15-s − 2.88e5·17-s + 7.81e5·19-s − 6.33e5·21-s + 5.06e4·23-s + 4.22e6·25-s − 9.46e6·27-s − 7.17e6·29-s + 4.31e6·31-s + 1.07e7·33-s + 1.47e7·35-s − 1.56e7·37-s − 2.97e7·39-s − 2.51e7·41-s + 7.51e7·43-s + 1.38e8·45-s + 8.37e7·47-s − 1.02e8·49-s − 4.46e7·51-s − 1.38e8·53-s − 2.50e8·55-s + ⋯
L(s)  = 1  + 1.10·3-s − 2.58·5-s − 0.643·7-s − 1.94·9-s + 1.42·11-s − 1.86·13-s − 2.85·15-s − 0.836·17-s + 1.37·19-s − 0.710·21-s + 0.0377·23-s + 2.16·25-s − 3.42·27-s − 1.88·29-s + 0.838·31-s + 1.57·33-s + 1.66·35-s − 1.37·37-s − 2.05·39-s − 1.38·41-s + 3.35·43-s + 5.01·45-s + 2.50·47-s − 2.53·49-s − 0.924·51-s − 2.40·53-s − 3.68·55-s + ⋯

Functional equation

Λ(s)=((224⋅196)s/2 ΓC(s)6 L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}
Λ(s)=((224⋅196)s/2 ΓC(s+9/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 224⋅1962^{24} \cdot 19^{6}
Sign: 11
Analytic conductor: 1.47321×10131.47321\times 10^{13}
Root analytic conductor: 12.512812.5128
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 224⋅196, ( :[9/2]6), 1)(12,\ 2^{24} \cdot 19^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
19 (1−p4T)6 ( 1 - p^{4} T )^{6}
good3 1−155T+62249T2−6105562T3+636046237pT4−19386269207p2T5+1680430278122p3T6−19386269207p11T7+636046237p19T8−6105562p27T9+62249p36T10−155p45T11+p54T12 1 - 155 T + 62249 T^{2} - 6105562 T^{3} + 636046237 p T^{4} - 19386269207 p^{2} T^{5} + 1680430278122 p^{3} T^{6} - 19386269207 p^{11} T^{7} + 636046237 p^{19} T^{8} - 6105562 p^{27} T^{9} + 62249 p^{36} T^{10} - 155 p^{45} T^{11} + p^{54} T^{12}
5 1+3612T+8819203T2+15226118454T3+4491105434991pT4+1174269510249486p2T5+335172475640149498p3T6+1174269510249486p11T7+4491105434991p19T8+15226118454p27T9+8819203p36T10+3612p45T11+p54T12 1 + 3612 T + 8819203 T^{2} + 15226118454 T^{3} + 4491105434991 p T^{4} + 1174269510249486 p^{2} T^{5} + 335172475640149498 p^{3} T^{6} + 1174269510249486 p^{11} T^{7} + 4491105434991 p^{19} T^{8} + 15226118454 p^{27} T^{9} + 8819203 p^{36} T^{10} + 3612 p^{45} T^{11} + p^{54} T^{12}
7 1+4085T+16985632pT2+12594232429p2T3+20742144301696p3T4+16781313847414901p4T5+18718742091583832402p5T6+16781313847414901p13T7+20742144301696p21T8+12594232429p29T9+16985632p37T10+4085p45T11+p54T12 1 + 4085 T + 16985632 p T^{2} + 12594232429 p^{2} T^{3} + 20742144301696 p^{3} T^{4} + 16781313847414901 p^{4} T^{5} + 18718742091583832402 p^{5} T^{6} + 16781313847414901 p^{13} T^{7} + 20742144301696 p^{21} T^{8} + 12594232429 p^{29} T^{9} + 16985632 p^{37} T^{10} + 4085 p^{45} T^{11} + p^{54} T^{12}
11 1−69312T+8919448955T2−362406965864970T3+30850521292803957655T4− 1 - 69312 T + 8919448955 T^{2} - 362406965864970 T^{3} + 30850521292803957655 T^{4} - 90 ⁣⋯ ⁣8290\!\cdots\!82T5+ T^{5} + 76 ⁣⋯ ⁣3076\!\cdots\!30T6− T^{6} - 90 ⁣⋯ ⁣8290\!\cdots\!82p9T7+30850521292803957655p18T8−362406965864970p27T9+8919448955p36T10−69312p45T11+p54T12 p^{9} T^{7} + 30850521292803957655 p^{18} T^{8} - 362406965864970 p^{27} T^{9} + 8919448955 p^{36} T^{10} - 69312 p^{45} T^{11} + p^{54} T^{12}
13 1+191747T+63256423831T2+8493310736420104T3+ 1 + 191747 T + 63256423831 T^{2} + 8493310736420104 T^{3} + 16 ⁣⋯ ⁣0516\!\cdots\!05T4+ T^{4} + 16 ⁣⋯ ⁣6916\!\cdots\!69T5+ T^{5} + 22 ⁣⋯ ⁣6222\!\cdots\!62T6+ T^{6} + 16 ⁣⋯ ⁣6916\!\cdots\!69p9T7+ p^{9} T^{7} + 16 ⁣⋯ ⁣0516\!\cdots\!05p18T8+8493310736420104p27T9+63256423831p36T10+191747p45T11+p54T12 p^{18} T^{8} + 8493310736420104 p^{27} T^{9} + 63256423831 p^{36} T^{10} + 191747 p^{45} T^{11} + p^{54} T^{12}
17 1+288195T+584693721722T2+128890679515643145T3+ 1 + 288195 T + 584693721722 T^{2} + 128890679515643145 T^{3} + 88 ⁣⋯ ⁣7688\!\cdots\!76pT4+ p T^{4} + 26 ⁣⋯ ⁣0726\!\cdots\!07T5+ T^{5} + 22 ⁣⋯ ⁣2022\!\cdots\!20T6+ T^{6} + 26 ⁣⋯ ⁣0726\!\cdots\!07p9T7+ p^{9} T^{7} + 88 ⁣⋯ ⁣7688\!\cdots\!76p19T8+128890679515643145p27T9+584693721722p36T10+288195p45T11+p54T12 p^{19} T^{8} + 128890679515643145 p^{27} T^{9} + 584693721722 p^{36} T^{10} + 288195 p^{45} T^{11} + p^{54} T^{12}
23 1−50697T+7848341516961T2−2213517228317023194T3+ 1 - 50697 T + 7848341516961 T^{2} - 2213517228317023194 T^{3} + 28 ⁣⋯ ⁣2328\!\cdots\!23T4− T^{4} - 10 ⁣⋯ ⁣7310\!\cdots\!73T5+ T^{5} + 61 ⁣⋯ ⁣4661\!\cdots\!46T6− T^{6} - 10 ⁣⋯ ⁣7310\!\cdots\!73p9T7+ p^{9} T^{7} + 28 ⁣⋯ ⁣2328\!\cdots\!23p18T8−2213517228317023194p27T9+7848341516961p36T10−50697p45T11+p54T12 p^{18} T^{8} - 2213517228317023194 p^{27} T^{9} + 7848341516961 p^{36} T^{10} - 50697 p^{45} T^{11} + p^{54} T^{12}
29 1+7178667T+44233248186399T2+ 1 + 7178667 T + 44233248186399 T^{2} + 16 ⁣⋯ ⁣6816\!\cdots\!68T3+ T^{3} + 62 ⁣⋯ ⁣8162\!\cdots\!81T4+ T^{4} + 12 ⁣⋯ ⁣8912\!\cdots\!89T5+ T^{5} + 43 ⁣⋯ ⁣5843\!\cdots\!58T6+ T^{6} + 12 ⁣⋯ ⁣8912\!\cdots\!89p9T7+ p^{9} T^{7} + 62 ⁣⋯ ⁣8162\!\cdots\!81p18T8+ p^{18} T^{8} + 16 ⁣⋯ ⁣6816\!\cdots\!68p27T9+44233248186399p36T10+7178667p45T11+p54T12 p^{27} T^{9} + 44233248186399 p^{36} T^{10} + 7178667 p^{45} T^{11} + p^{54} T^{12}
31 1−4310716T+91495602933162T2− 1 - 4310716 T + 91495602933162 T^{2} - 39 ⁣⋯ ⁣7639\!\cdots\!76T3+ T^{3} + 48 ⁣⋯ ⁣5548\!\cdots\!55T4− T^{4} - 17 ⁣⋯ ⁣2017\!\cdots\!20T5+ T^{5} + 15 ⁣⋯ ⁣0015\!\cdots\!00T6− T^{6} - 17 ⁣⋯ ⁣2017\!\cdots\!20p9T7+ p^{9} T^{7} + 48 ⁣⋯ ⁣5548\!\cdots\!55p18T8− p^{18} T^{8} - 39 ⁣⋯ ⁣7639\!\cdots\!76p27T9+91495602933162p36T10−4310716p45T11+p54T12 p^{27} T^{9} + 91495602933162 p^{36} T^{10} - 4310716 p^{45} T^{11} + p^{54} T^{12}
37 1+15687040T+477200595260658T2+ 1 + 15687040 T + 477200595260658 T^{2} + 41 ⁣⋯ ⁣1241\!\cdots\!12T3+ T^{3} + 77 ⁣⋯ ⁣9177\!\cdots\!91T4+ T^{4} + 42 ⁣⋯ ⁣6842\!\cdots\!68T5+ T^{5} + 89 ⁣⋯ ⁣1289\!\cdots\!12T6+ T^{6} + 42 ⁣⋯ ⁣6842\!\cdots\!68p9T7+ p^{9} T^{7} + 77 ⁣⋯ ⁣9177\!\cdots\!91p18T8+ p^{18} T^{8} + 41 ⁣⋯ ⁣1241\!\cdots\!12p27T9+477200595260658p36T10+15687040p45T11+p54T12 p^{27} T^{9} + 477200595260658 p^{36} T^{10} + 15687040 p^{45} T^{11} + p^{54} T^{12}
41 1+25134306T+1079097703143394T2+ 1 + 25134306 T + 1079097703143394 T^{2} + 15 ⁣⋯ ⁣5015\!\cdots\!50T3+ T^{3} + 49 ⁣⋯ ⁣6749\!\cdots\!67T4+ T^{4} + 55 ⁣⋯ ⁣0055\!\cdots\!00T5+ T^{5} + 17 ⁣⋯ ⁣0417\!\cdots\!04T6+ T^{6} + 55 ⁣⋯ ⁣0055\!\cdots\!00p9T7+ p^{9} T^{7} + 49 ⁣⋯ ⁣6749\!\cdots\!67p18T8+ p^{18} T^{8} + 15 ⁣⋯ ⁣5015\!\cdots\!50p27T9+1079097703143394p36T10+25134306p45T11+p54T12 p^{27} T^{9} + 1079097703143394 p^{36} T^{10} + 25134306 p^{45} T^{11} + p^{54} T^{12}
43 1−75118674T+4425622451146347T2− 1 - 75118674 T + 4425622451146347 T^{2} - 16 ⁣⋯ ⁣0216\!\cdots\!02T3+ T^{3} + 56 ⁣⋯ ⁣5956\!\cdots\!59T4− T^{4} - 14 ⁣⋯ ⁣4414\!\cdots\!44T5+ T^{5} + 36 ⁣⋯ ⁣9836\!\cdots\!98T6− T^{6} - 14 ⁣⋯ ⁣4414\!\cdots\!44p9T7+ p^{9} T^{7} + 56 ⁣⋯ ⁣5956\!\cdots\!59p18T8− p^{18} T^{8} - 16 ⁣⋯ ⁣0216\!\cdots\!02p27T9+4425622451146347p36T10−75118674p45T11+p54T12 p^{27} T^{9} + 4425622451146347 p^{36} T^{10} - 75118674 p^{45} T^{11} + p^{54} T^{12}
47 1−83731938T+7369210577066907T2− 1 - 83731938 T + 7369210577066907 T^{2} - 39 ⁣⋯ ⁣3039\!\cdots\!30T3+ T^{3} + 21 ⁣⋯ ⁣4721\!\cdots\!47T4− T^{4} - 82 ⁣⋯ ⁣2482\!\cdots\!24T5+ T^{5} + 31 ⁣⋯ ⁣9031\!\cdots\!90T6− T^{6} - 82 ⁣⋯ ⁣2482\!\cdots\!24p9T7+ p^{9} T^{7} + 21 ⁣⋯ ⁣4721\!\cdots\!47p18T8− p^{18} T^{8} - 39 ⁣⋯ ⁣3039\!\cdots\!30p27T9+7369210577066907p36T10−83731938p45T11+p54T12 p^{27} T^{9} + 7369210577066907 p^{36} T^{10} - 83731938 p^{45} T^{11} + p^{54} T^{12}
53 1+138019203T+23264355593581535T2+ 1 + 138019203 T + 23264355593581535 T^{2} + 20 ⁣⋯ ⁣2020\!\cdots\!20T3+ T^{3} + 20 ⁣⋯ ⁣1320\!\cdots\!13T4+ T^{4} + 13 ⁣⋯ ⁣5313\!\cdots\!53T5+ T^{5} + 89 ⁣⋯ ⁣4689\!\cdots\!46T6+ T^{6} + 13 ⁣⋯ ⁣5313\!\cdots\!53p9T7+ p^{9} T^{7} + 20 ⁣⋯ ⁣1320\!\cdots\!13p18T8+ p^{18} T^{8} + 20 ⁣⋯ ⁣2020\!\cdots\!20p27T9+23264355593581535p36T10+138019203p45T11+p54T12 p^{27} T^{9} + 23264355593581535 p^{36} T^{10} + 138019203 p^{45} T^{11} + p^{54} T^{12}
59 1−7809915T+42493160832721529T2− 1 - 7809915 T + 42493160832721529 T^{2} - 13 ⁣⋯ ⁣1813\!\cdots\!18T3+ T^{3} + 81 ⁣⋯ ⁣1181\!\cdots\!11T4− T^{4} - 10 ⁣⋯ ⁣4710\!\cdots\!47T5+ T^{5} + 90 ⁣⋯ ⁣3890\!\cdots\!38T6− T^{6} - 10 ⁣⋯ ⁣4710\!\cdots\!47p9T7+ p^{9} T^{7} + 81 ⁣⋯ ⁣1181\!\cdots\!11p18T8− p^{18} T^{8} - 13 ⁣⋯ ⁣1813\!\cdots\!18p27T9+42493160832721529p36T10−7809915p45T11+p54T12 p^{27} T^{9} + 42493160832721529 p^{36} T^{10} - 7809915 p^{45} T^{11} + p^{54} T^{12}
61 1−191946566T+54780670024444367T2− 1 - 191946566 T + 54780670024444367 T^{2} - 82 ⁣⋯ ⁣9082\!\cdots\!90T3+ T^{3} + 13 ⁣⋯ ⁣8713\!\cdots\!87T4− T^{4} - 16 ⁣⋯ ⁣6416\!\cdots\!64T5+ T^{5} + 20 ⁣⋯ ⁣7020\!\cdots\!70T6− T^{6} - 16 ⁣⋯ ⁣6416\!\cdots\!64p9T7+ p^{9} T^{7} + 13 ⁣⋯ ⁣8713\!\cdots\!87p18T8− p^{18} T^{8} - 82 ⁣⋯ ⁣9082\!\cdots\!90p27T9+54780670024444367p36T10−191946566p45T11+p54T12 p^{27} T^{9} + 54780670024444367 p^{36} T^{10} - 191946566 p^{45} T^{11} + p^{54} T^{12}
67 1−16109787T+108574986848183601T2− 1 - 16109787 T + 108574986848183601 T^{2} - 72 ⁣⋯ ⁣5072\!\cdots\!50T3+ T^{3} + 54 ⁣⋯ ⁣4354\!\cdots\!43T4− T^{4} - 92 ⁣⋯ ⁣5192\!\cdots\!51T5+ T^{5} + 17 ⁣⋯ ⁣1417\!\cdots\!14T6− T^{6} - 92 ⁣⋯ ⁣5192\!\cdots\!51p9T7+ p^{9} T^{7} + 54 ⁣⋯ ⁣4354\!\cdots\!43p18T8− p^{18} T^{8} - 72 ⁣⋯ ⁣5072\!\cdots\!50p27T9+108574986848183601p36T10−16109787p45T11+p54T12 p^{27} T^{9} + 108574986848183601 p^{36} T^{10} - 16109787 p^{45} T^{11} + p^{54} T^{12}
71 1−264469698T+176605075037501922T2− 1 - 264469698 T + 176605075037501922 T^{2} - 32 ⁣⋯ ⁣7432\!\cdots\!74T3+ T^{3} + 11 ⁣⋯ ⁣5511\!\cdots\!55T4− T^{4} - 16 ⁣⋯ ⁣1216\!\cdots\!12T5+ T^{5} + 47 ⁣⋯ ⁣3647\!\cdots\!36T6− T^{6} - 16 ⁣⋯ ⁣1216\!\cdots\!12p9T7+ p^{9} T^{7} + 11 ⁣⋯ ⁣5511\!\cdots\!55p18T8− p^{18} T^{8} - 32 ⁣⋯ ⁣7432\!\cdots\!74p27T9+176605075037501922p36T10−264469698p45T11+p54T12 p^{27} T^{9} + 176605075037501922 p^{36} T^{10} - 264469698 p^{45} T^{11} + p^{54} T^{12}
73 1+287572857T+189879359499479490T2+ 1 + 287572857 T + 189879359499479490 T^{2} + 45 ⁣⋯ ⁣9145\!\cdots\!91T3+ T^{3} + 19 ⁣⋯ ⁣2019\!\cdots\!20T4+ T^{4} + 39 ⁣⋯ ⁣4939\!\cdots\!49T5+ T^{5} + 12 ⁣⋯ ⁣4812\!\cdots\!48T6+ T^{6} + 39 ⁣⋯ ⁣4939\!\cdots\!49p9T7+ p^{9} T^{7} + 19 ⁣⋯ ⁣2019\!\cdots\!20p18T8+ p^{18} T^{8} + 45 ⁣⋯ ⁣9145\!\cdots\!91p27T9+189879359499479490p36T10+287572857p45T11+p54T12 p^{27} T^{9} + 189879359499479490 p^{36} T^{10} + 287572857 p^{45} T^{11} + p^{54} T^{12}
79 1−86534002T+338621828328018454T2− 1 - 86534002 T + 338621828328018454 T^{2} - 30 ⁣⋯ ⁣9430\!\cdots\!94T3+ T^{3} + 40 ⁣⋯ ⁣3140\!\cdots\!31T4+ T^{4} + 50 ⁣⋯ ⁣4850\!\cdots\!48T5+ T^{5} + 35 ⁣⋯ ⁣2835\!\cdots\!28T6+ T^{6} + 50 ⁣⋯ ⁣4850\!\cdots\!48p9T7+ p^{9} T^{7} + 40 ⁣⋯ ⁣3140\!\cdots\!31p18T8− p^{18} T^{8} - 30 ⁣⋯ ⁣9430\!\cdots\!94p27T9+338621828328018454p36T10−86534002p45T11+p54T12 p^{27} T^{9} + 338621828328018454 p^{36} T^{10} - 86534002 p^{45} T^{11} + p^{54} T^{12}
83 1−359214570T+415846672018866310T2− 1 - 359214570 T + 415846672018866310 T^{2} - 17 ⁣⋯ ⁣1817\!\cdots\!18T3+ T^{3} + 16 ⁣⋯ ⁣5116\!\cdots\!51T4− T^{4} - 52 ⁣⋯ ⁣5652\!\cdots\!56T5+ T^{5} + 32 ⁣⋯ ⁣3232\!\cdots\!32T6− T^{6} - 52 ⁣⋯ ⁣5652\!\cdots\!56p9T7+ p^{9} T^{7} + 16 ⁣⋯ ⁣5116\!\cdots\!51p18T8− p^{18} T^{8} - 17 ⁣⋯ ⁣1817\!\cdots\!18p27T9+415846672018866310p36T10−359214570p45T11+p54T12 p^{27} T^{9} + 415846672018866310 p^{36} T^{10} - 359214570 p^{45} T^{11} + p^{54} T^{12}
89 1+2263866306T+3767251604053176142T2+ 1 + 2263866306 T + 3767251604053176142 T^{2} + 43 ⁣⋯ ⁣5843\!\cdots\!58T3+ T^{3} + 41 ⁣⋯ ⁣1941\!\cdots\!19T4+ T^{4} + 31 ⁣⋯ ⁣0431\!\cdots\!04T5+ T^{5} + 20 ⁣⋯ ⁣1620\!\cdots\!16T6+ T^{6} + 31 ⁣⋯ ⁣0431\!\cdots\!04p9T7+ p^{9} T^{7} + 41 ⁣⋯ ⁣1941\!\cdots\!19p18T8+ p^{18} T^{8} + 43 ⁣⋯ ⁣5843\!\cdots\!58p27T9+3767251604053176142p36T10+2263866306p45T11+p54T12 p^{27} T^{9} + 3767251604053176142 p^{36} T^{10} + 2263866306 p^{45} T^{11} + p^{54} T^{12}
97 1−2705893460T+5799960519982386998T2− 1 - 2705893460 T + 5799960519982386998 T^{2} - 86 ⁣⋯ ⁣3686\!\cdots\!36T3+ T^{3} + 11 ⁣⋯ ⁣4311\!\cdots\!43T4− T^{4} - 12 ⁣⋯ ⁣4812\!\cdots\!48T5+ T^{5} + 11 ⁣⋯ ⁣5211\!\cdots\!52T6− T^{6} - 12 ⁣⋯ ⁣4812\!\cdots\!48p9T7+ p^{9} T^{7} + 11 ⁣⋯ ⁣4311\!\cdots\!43p18T8− p^{18} T^{8} - 86 ⁣⋯ ⁣3686\!\cdots\!36p27T9+5799960519982386998p36T10−2705893460p45T11+p54T12 p^{27} T^{9} + 5799960519982386998 p^{36} T^{10} - 2705893460 p^{45} T^{11} + p^{54} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.32606015785312617099435563727, −4.97439979371339355762440700549, −4.90743031563687052132025268645, −4.76819836527642734393239583949, −4.69831489565144513795432449396, −4.28017901591967880003067367610, −4.09826571087795917041491060977, −3.99363191227807002934032800982, −3.76195071122511102693228578783, −3.57105788946479838788758252205, −3.52169047662600886328513999728, −3.47366415324261029659972736265, −3.28790831304999798092216640025, −2.84560355705821983463752193140, −2.80836092845989985703871622713, −2.48168396537269024744054215630, −2.37521642072884107727010256541, −2.34198832648923824751133753948, −2.29274393113782652630247831485, −1.73130648479387067343484345335, −1.40539078003259051662459946193, −1.36189063721696545706570388715, −1.14173151201810702730655083668, −0.866866338636338954942571721172, −0.74538495576022260711615726037, 0, 0, 0, 0, 0, 0, 0.74538495576022260711615726037, 0.866866338636338954942571721172, 1.14173151201810702730655083668, 1.36189063721696545706570388715, 1.40539078003259051662459946193, 1.73130648479387067343484345335, 2.29274393113782652630247831485, 2.34198832648923824751133753948, 2.37521642072884107727010256541, 2.48168396537269024744054215630, 2.80836092845989985703871622713, 2.84560355705821983463752193140, 3.28790831304999798092216640025, 3.47366415324261029659972736265, 3.52169047662600886328513999728, 3.57105788946479838788758252205, 3.76195071122511102693228578783, 3.99363191227807002934032800982, 4.09826571087795917041491060977, 4.28017901591967880003067367610, 4.69831489565144513795432449396, 4.76819836527642734393239583949, 4.90743031563687052132025268645, 4.97439979371339355762440700549, 5.32606015785312617099435563727

Graph of the ZZ-function along the critical line

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