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Properties

Label 36-640e18-1.1-c3e18-0-0
Degree 3636
Conductor 3.245×10503.245\times 10^{50}
Sign 11
Analytic cond. 2.43685×10282.43685\times 10^{28}
Root an. cond. 6.145016.14501
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 162·9-s − 28·11-s − 68·19-s − 9·25-s − 340·29-s + 336·31-s + 236·41-s − 324·45-s + 2.64e3·49-s + 56·55-s − 1.75e3·59-s + 780·61-s + 72·71-s − 1.95e3·79-s + 1.25e4·81-s − 292·89-s + 136·95-s − 4.53e3·99-s + 3.72e3·101-s − 1.74e3·109-s − 1.06e4·121-s − 452·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.178·5-s + 6·9-s − 0.767·11-s − 0.821·19-s − 0.0719·25-s − 2.17·29-s + 1.94·31-s + 0.898·41-s − 1.07·45-s + 54/7·49-s + 0.137·55-s − 3.87·59-s + 1.63·61-s + 0.120·71-s − 2.77·79-s + 17.2·81-s − 0.347·89-s + 0.146·95-s − 4.60·99-s + 3.66·101-s − 1.53·109-s − 8.01·121-s − 0.323·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

Λ(s)=((2126⋅518)s/2 ΓC(s)18 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{126} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((2126⋅518)s/2 ΓC(s+3/2)18 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{126} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3636
Conductor: 2126⋅5182^{126} \cdot 5^{18}
Sign: 11
Analytic conductor: 2.43685×10282.43685\times 10^{28}
Root analytic conductor: 6.145016.14501
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (36, 2126⋅518, ( :[3/2]18), 1)(36,\ 2^{126} \cdot 5^{18} ,\ ( \ : [3/2]^{18} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.062844807920.06284480792
L(12)L(\frac12) ≈\approx 0.062844807920.06284480792
L(52)L(\frac{5}{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
5 1+2T+13T2+496T3+10556T4−77448T5+521596pT6+85136p2T7+676718p3T8−1895412p4T9+676718p6T10+85136p8T11+521596p10T12−77448p12T13+10556p15T14+496p18T15+13p21T16+2p24T17+p27T18 1 + 2 T + 13 T^{2} + 496 T^{3} + 10556 T^{4} - 77448 T^{5} + 521596 p T^{6} + 85136 p^{2} T^{7} + 676718 p^{3} T^{8} - 1895412 p^{4} T^{9} + 676718 p^{6} T^{10} + 85136 p^{8} T^{11} + 521596 p^{10} T^{12} - 77448 p^{12} T^{13} + 10556 p^{15} T^{14} + 496 p^{18} T^{15} + 13 p^{21} T^{16} + 2 p^{24} T^{17} + p^{27} T^{18}
good3 1−2p4T2+13697T4−89056p2T6+11917900pT8−1276761368T10+12599676220pT12−321794417120pT14+23009371354142T16−581214078652172T18+23009371354142p6T20−321794417120p13T22+12599676220p19T24−1276761368p24T26+11917900p31T28−89056p38T30+13697p42T32−2p52T34+p54T36 1 - 2 p^{4} T^{2} + 13697 T^{4} - 89056 p^{2} T^{6} + 11917900 p T^{8} - 1276761368 T^{10} + 12599676220 p T^{12} - 321794417120 p T^{14} + 23009371354142 T^{16} - 581214078652172 T^{18} + 23009371354142 p^{6} T^{20} - 321794417120 p^{13} T^{22} + 12599676220 p^{19} T^{24} - 1276761368 p^{24} T^{26} + 11917900 p^{31} T^{28} - 89056 p^{38} T^{30} + 13697 p^{42} T^{32} - 2 p^{52} T^{34} + p^{54} T^{36}
7 1−54p2T2+3505273T4−3175456896T6+2244628262820T8−1321783876957640T10+671383044235627732T12− 1 - 54 p^{2} T^{2} + 3505273 T^{4} - 3175456896 T^{6} + 2244628262820 T^{8} - 1321783876957640 T^{10} + 671383044235627732 T^{12} - 30 ⁣⋯ ⁣0030\!\cdots\!00T14+ T^{14} + 12 ⁣⋯ ⁣3012\!\cdots\!30T16− T^{16} - 43 ⁣⋯ ⁣7643\!\cdots\!76T18+ T^{18} + 12 ⁣⋯ ⁣3012\!\cdots\!30p6T20− p^{6} T^{20} - 30 ⁣⋯ ⁣0030\!\cdots\!00p12T22+671383044235627732p18T24−1321783876957640p24T26+2244628262820p30T28−3175456896p36T30+3505273p42T32−54p50T34+p54T36 p^{12} T^{22} + 671383044235627732 p^{18} T^{24} - 1321783876957640 p^{24} T^{26} + 2244628262820 p^{30} T^{28} - 3175456896 p^{36} T^{30} + 3505273 p^{42} T^{32} - 54 p^{50} T^{34} + p^{54} T^{36}
11 (1+14T+5627T2+3120pT3+15719060T4+15589128T5+32297994924T6+11904627440T7+4972487740586pT8+52083142943380T9+4972487740586p4T10+11904627440p6T11+32297994924p9T12+15589128p12T13+15719060p15T14+3120p19T15+5627p21T16+14p24T17+p27T18)2 ( 1 + 14 T + 5627 T^{2} + 3120 p T^{3} + 15719060 T^{4} + 15589128 T^{5} + 32297994924 T^{6} + 11904627440 T^{7} + 4972487740586 p T^{8} + 52083142943380 T^{9} + 4972487740586 p^{4} T^{10} + 11904627440 p^{6} T^{11} + 32297994924 p^{9} T^{12} + 15589128 p^{12} T^{13} + 15719060 p^{15} T^{14} + 3120 p^{19} T^{15} + 5627 p^{21} T^{16} + 14 p^{24} T^{17} + p^{27} T^{18} )^{2}
13 1−18410T2+168206929T4−1021303084400T6+4682432877868276T8−17526512779132279448T10+ 1 - 18410 T^{2} + 168206929 T^{4} - 1021303084400 T^{6} + 4682432877868276 T^{8} - 17526512779132279448 T^{10} + 56 ⁣⋯ ⁣0456\!\cdots\!04T12− T^{12} - 16 ⁣⋯ ⁣5216\!\cdots\!52T14+ T^{14} + 41 ⁣⋯ ⁣0641\!\cdots\!06T16− T^{16} - 96 ⁣⋯ ⁣6896\!\cdots\!68T18+ T^{18} + 41 ⁣⋯ ⁣0641\!\cdots\!06p6T20− p^{6} T^{20} - 16 ⁣⋯ ⁣5216\!\cdots\!52p12T22+ p^{12} T^{22} + 56 ⁣⋯ ⁣0456\!\cdots\!04p18T24−17526512779132279448p24T26+4682432877868276p30T28−1021303084400p36T30+168206929p42T32−18410p48T34+p54T36 p^{18} T^{24} - 17526512779132279448 p^{24} T^{26} + 4682432877868276 p^{30} T^{28} - 1021303084400 p^{36} T^{30} + 168206929 p^{42} T^{32} - 18410 p^{48} T^{34} + p^{54} T^{36}
17 1−45282T2+1080021241T4−17672305163440T6+220352751485334260T8− 1 - 45282 T^{2} + 1080021241 T^{4} - 17672305163440 T^{6} + 220352751485334260 T^{8} - 22 ⁣⋯ ⁣9622\!\cdots\!96T10+ T^{10} + 18 ⁣⋯ ⁣4018\!\cdots\!40T12− T^{12} - 13 ⁣⋯ ⁣4413\!\cdots\!44T14+ T^{14} + 79 ⁣⋯ ⁣7479\!\cdots\!74T16− T^{16} - 42 ⁣⋯ ⁣3642\!\cdots\!36T18+ T^{18} + 79 ⁣⋯ ⁣7479\!\cdots\!74p6T20− p^{6} T^{20} - 13 ⁣⋯ ⁣4413\!\cdots\!44p12T22+ p^{12} T^{22} + 18 ⁣⋯ ⁣4018\!\cdots\!40p18T24− p^{18} T^{24} - 22 ⁣⋯ ⁣9622\!\cdots\!96p24T26+220352751485334260p30T28−17672305163440p36T30+1080021241p42T32−45282p48T34+p54T36 p^{24} T^{26} + 220352751485334260 p^{30} T^{28} - 17672305163440 p^{36} T^{30} + 1080021241 p^{42} T^{32} - 45282 p^{48} T^{34} + p^{54} T^{36}
19 (1+34T+31907T2+1025104T3+521944740T4+15342161400T5+5760096801308T6+151273640580272T7+48681526084193758T8+1148710323738341260T9+48681526084193758p3T10+151273640580272p6T11+5760096801308p9T12+15342161400p12T13+521944740p15T14+1025104p18T15+31907p21T16+34p24T17+p27T18)2 ( 1 + 34 T + 31907 T^{2} + 1025104 T^{3} + 521944740 T^{4} + 15342161400 T^{5} + 5760096801308 T^{6} + 151273640580272 T^{7} + 48681526084193758 T^{8} + 1148710323738341260 T^{9} + 48681526084193758 p^{3} T^{10} + 151273640580272 p^{6} T^{11} + 5760096801308 p^{9} T^{12} + 15342161400 p^{12} T^{13} + 521944740 p^{15} T^{14} + 1025104 p^{18} T^{15} + 31907 p^{21} T^{16} + 34 p^{24} T^{17} + p^{27} T^{18} )^{2}
23 1−110982T2+269302639pT4−229175593427776T6+6286529227274876740T8− 1 - 110982 T^{2} + 269302639 p T^{4} - 229175593427776 T^{6} + 6286529227274876740 T^{8} - 13 ⁣⋯ ⁣1213\!\cdots\!12T10+ T^{10} + 24 ⁣⋯ ⁣7224\!\cdots\!72T12− T^{12} - 37 ⁣⋯ ⁣3637\!\cdots\!36T14+ T^{14} + 51 ⁣⋯ ⁣0651\!\cdots\!06T16− T^{16} - 64 ⁣⋯ ⁣6864\!\cdots\!68T18+ T^{18} + 51 ⁣⋯ ⁣0651\!\cdots\!06p6T20− p^{6} T^{20} - 37 ⁣⋯ ⁣3637\!\cdots\!36p12T22+ p^{12} T^{22} + 24 ⁣⋯ ⁣7224\!\cdots\!72p18T24− p^{18} T^{24} - 13 ⁣⋯ ⁣1213\!\cdots\!12p24T26+6286529227274876740p30T28−229175593427776p36T30+269302639p43T32−110982p48T34+p54T36 p^{24} T^{26} + 6286529227274876740 p^{30} T^{28} - 229175593427776 p^{36} T^{30} + 269302639 p^{43} T^{32} - 110982 p^{48} T^{34} + p^{54} T^{36}
29 (1+170T+107517T2+13932208T3+5725536020T4+21104110392pT5+214303413402420T6+19821854179339344T7+6349836904888612174T8+ ( 1 + 170 T + 107517 T^{2} + 13932208 T^{3} + 5725536020 T^{4} + 21104110392 p T^{5} + 214303413402420 T^{6} + 19821854179339344 T^{7} + 6349836904888612174 T^{8} + 52 ⁣⋯ ⁣6052\!\cdots\!60T9+6349836904888612174p3T10+19821854179339344p6T11+214303413402420p9T12+21104110392p13T13+5725536020p15T14+13932208p18T15+107517p21T16+170p24T17+p27T18)2 T^{9} + 6349836904888612174 p^{3} T^{10} + 19821854179339344 p^{6} T^{11} + 214303413402420 p^{9} T^{12} + 21104110392 p^{13} T^{13} + 5725536020 p^{15} T^{14} + 13932208 p^{18} T^{15} + 107517 p^{21} T^{16} + 170 p^{24} T^{17} + p^{27} T^{18} )^{2}
31 (1−168T+132023T2−22283328T3+10383373956T4−1488537215328T5+533181488401932T6−69306017242405312T7+20577323736403541598T8− ( 1 - 168 T + 132023 T^{2} - 22283328 T^{3} + 10383373956 T^{4} - 1488537215328 T^{5} + 533181488401932 T^{6} - 69306017242405312 T^{7} + 20577323736403541598 T^{8} - 23 ⁣⋯ ⁣2823\!\cdots\!28T9+20577323736403541598p3T10−69306017242405312p6T11+533181488401932p9T12−1488537215328p12T13+10383373956p15T14−22283328p18T15+132023p21T16−168p24T17+p27T18)2 T^{9} + 20577323736403541598 p^{3} T^{10} - 69306017242405312 p^{6} T^{11} + 533181488401932 p^{9} T^{12} - 1488537215328 p^{12} T^{13} + 10383373956 p^{15} T^{14} - 22283328 p^{18} T^{15} + 132023 p^{21} T^{16} - 168 p^{24} T^{17} + p^{27} T^{18} )^{2}
37 1−351370T2+1777230925pT4−8668242846445296T6+ 1 - 351370 T^{2} + 1777230925 p T^{4} - 8668242846445296 T^{6} + 90 ⁣⋯ ⁣3690\!\cdots\!36T8− T^{8} - 79 ⁣⋯ ⁣8479\!\cdots\!84T10+ T^{10} + 60 ⁣⋯ ⁣2860\!\cdots\!28T12− T^{12} - 40 ⁣⋯ ⁣8040\!\cdots\!80T14+ T^{14} + 24 ⁣⋯ ⁣8224\!\cdots\!82T16− T^{16} - 12 ⁣⋯ ⁣6012\!\cdots\!60T18+ T^{18} + 24 ⁣⋯ ⁣8224\!\cdots\!82p6T20− p^{6} T^{20} - 40 ⁣⋯ ⁣8040\!\cdots\!80p12T22+ p^{12} T^{22} + 60 ⁣⋯ ⁣2860\!\cdots\!28p18T24− p^{18} T^{24} - 79 ⁣⋯ ⁣8479\!\cdots\!84p24T26+ p^{24} T^{26} + 90 ⁣⋯ ⁣3690\!\cdots\!36p30T28−8668242846445296p36T30+1777230925p43T32−351370p48T34+p54T36 p^{30} T^{28} - 8668242846445296 p^{36} T^{30} + 1777230925 p^{43} T^{32} - 351370 p^{48} T^{34} + p^{54} T^{36}
41 (1−118T+306417T2−18849056T3+42700765332T4+723627236344T5+3540593544233028T6+441956777709999008T7+ ( 1 - 118 T + 306417 T^{2} - 18849056 T^{3} + 42700765332 T^{4} + 723627236344 T^{5} + 3540593544233028 T^{6} + 441956777709999008 T^{7} + 22 ⁣⋯ ⁣2222\!\cdots\!22T8+ T^{8} + 46 ⁣⋯ ⁣8446\!\cdots\!84T9+ T^{9} + 22 ⁣⋯ ⁣2222\!\cdots\!22p3T10+441956777709999008p6T11+3540593544233028p9T12+723627236344p12T13+42700765332p15T14−18849056p18T15+306417p21T16−118p24T17+p27T18)2 p^{3} T^{10} + 441956777709999008 p^{6} T^{11} + 3540593544233028 p^{9} T^{12} + 723627236344 p^{12} T^{13} + 42700765332 p^{15} T^{14} - 18849056 p^{18} T^{15} + 306417 p^{21} T^{16} - 118 p^{24} T^{17} + p^{27} T^{18} )^{2}
43 1−817186T2+316383232561T4−77191645858921760T6+ 1 - 817186 T^{2} + 316383232561 T^{4} - 77191645858921760 T^{6} + 13 ⁣⋯ ⁣3613\!\cdots\!36T8− T^{8} - 40 ⁣⋯ ⁣0040\!\cdots\!00pT10+ p T^{10} + 17 ⁣⋯ ⁣3617\!\cdots\!36T12− T^{12} - 14 ⁣⋯ ⁣8414\!\cdots\!84T14+ T^{14} + 10 ⁣⋯ ⁣2610\!\cdots\!26T16− T^{16} - 79 ⁣⋯ ⁣4079\!\cdots\!40T18+ T^{18} + 10 ⁣⋯ ⁣2610\!\cdots\!26p6T20− p^{6} T^{20} - 14 ⁣⋯ ⁣8414\!\cdots\!84p12T22+ p^{12} T^{22} + 17 ⁣⋯ ⁣3617\!\cdots\!36p18T24− p^{18} T^{24} - 40 ⁣⋯ ⁣0040\!\cdots\!00p25T26+ p^{25} T^{26} + 13 ⁣⋯ ⁣3613\!\cdots\!36p30T28−77191645858921760p36T30+316383232561p42T32−817186p48T34+p54T36 p^{30} T^{28} - 77191645858921760 p^{36} T^{30} + 316383232561 p^{42} T^{32} - 817186 p^{48} T^{34} + p^{54} T^{36}
47 1−976438T2+463777655113T4−142404872157421440T6+ 1 - 976438 T^{2} + 463777655113 T^{4} - 142404872157421440 T^{6} + 31 ⁣⋯ ⁣1231\!\cdots\!12T8− T^{8} - 54 ⁣⋯ ⁣5254\!\cdots\!52T10+ T^{10} + 76 ⁣⋯ ⁣4476\!\cdots\!44T12− T^{12} - 91 ⁣⋯ ⁣2491\!\cdots\!24T14+ T^{14} + 99 ⁣⋯ ⁣3899\!\cdots\!38T16− T^{16} - 10 ⁣⋯ ⁣9210\!\cdots\!92T18+ T^{18} + 99 ⁣⋯ ⁣3899\!\cdots\!38p6T20− p^{6} T^{20} - 91 ⁣⋯ ⁣2491\!\cdots\!24p12T22+ p^{12} T^{22} + 76 ⁣⋯ ⁣4476\!\cdots\!44p18T24− p^{18} T^{24} - 54 ⁣⋯ ⁣5254\!\cdots\!52p24T26+ p^{24} T^{26} + 31 ⁣⋯ ⁣1231\!\cdots\!12p30T28−142404872157421440p36T30+463777655113p42T32−976438p48T34+p54T36 p^{30} T^{28} - 142404872157421440 p^{36} T^{30} + 463777655113 p^{42} T^{32} - 976438 p^{48} T^{34} + p^{54} T^{36}
53 1−1520026T2+1135748878529T4−560763301886123248T6+ 1 - 1520026 T^{2} + 1135748878529 T^{4} - 560763301886123248 T^{6} + 20 ⁣⋯ ⁣8020\!\cdots\!80T8− T^{8} - 61 ⁣⋯ ⁣7661\!\cdots\!76T10+ T^{10} + 15 ⁣⋯ ⁣0415\!\cdots\!04T12− T^{12} - 31 ⁣⋯ ⁣6831\!\cdots\!68T14+ T^{14} + 58 ⁣⋯ ⁣8658\!\cdots\!86T16− T^{16} - 92 ⁣⋯ ⁣4492\!\cdots\!44T18+ T^{18} + 58 ⁣⋯ ⁣8658\!\cdots\!86p6T20− p^{6} T^{20} - 31 ⁣⋯ ⁣6831\!\cdots\!68p12T22+ p^{12} T^{22} + 15 ⁣⋯ ⁣0415\!\cdots\!04p18T24− p^{18} T^{24} - 61 ⁣⋯ ⁣7661\!\cdots\!76p24T26+ p^{24} T^{26} + 20 ⁣⋯ ⁣8020\!\cdots\!80p30T28−560763301886123248p36T30+1135748878529p42T32−1520026p48T34+p54T36 p^{30} T^{28} - 560763301886123248 p^{36} T^{30} + 1135748878529 p^{42} T^{32} - 1520026 p^{48} T^{34} + p^{54} T^{36}
59 (1+878T+1236955T2+620099760T3+553755425508T4+199575505245768T5+172355456546221884T6+57782131388837538640T7+ ( 1 + 878 T + 1236955 T^{2} + 620099760 T^{3} + 553755425508 T^{4} + 199575505245768 T^{5} + 172355456546221884 T^{6} + 57782131388837538640 T^{7} + 47 ⁣⋯ ⁣1047\!\cdots\!10T8+ T^{8} + 14 ⁣⋯ ⁣8814\!\cdots\!88T9+ T^{9} + 47 ⁣⋯ ⁣1047\!\cdots\!10p3T10+57782131388837538640p6T11+172355456546221884p9T12+199575505245768p12T13+553755425508p15T14+620099760p18T15+1236955p21T16+878p24T17+p27T18)2 p^{3} T^{10} + 57782131388837538640 p^{6} T^{11} + 172355456546221884 p^{9} T^{12} + 199575505245768 p^{12} T^{13} + 553755425508 p^{15} T^{14} + 620099760 p^{18} T^{15} + 1236955 p^{21} T^{16} + 878 p^{24} T^{17} + p^{27} T^{18} )^{2}
61 (1−390T+946533T2−375530288T3+444635912140T4−173806478525608T5+143980830611353404T6−52074599746897680592T7+ ( 1 - 390 T + 946533 T^{2} - 375530288 T^{3} + 444635912140 T^{4} - 173806478525608 T^{5} + 143980830611353404 T^{6} - 52074599746897680592 T^{7} + 37 ⁣⋯ ⁣6237\!\cdots\!62T8− T^{8} - 12 ⁣⋯ ⁣2412\!\cdots\!24T9+ T^{9} + 37 ⁣⋯ ⁣6237\!\cdots\!62p3T10−52074599746897680592p6T11+143980830611353404p9T12−173806478525608p12T13+444635912140p15T14−375530288p18T15+946533p21T16−390p24T17+p27T18)2 p^{3} T^{10} - 52074599746897680592 p^{6} T^{11} + 143980830611353404 p^{9} T^{12} - 173806478525608 p^{12} T^{13} + 444635912140 p^{15} T^{14} - 375530288 p^{18} T^{15} + 946533 p^{21} T^{16} - 390 p^{24} T^{17} + p^{27} T^{18} )^{2}
67 1−2700338T2+3533629828065T4−2996289198536038944T6+ 1 - 2700338 T^{2} + 3533629828065 T^{4} - 2996289198536038944 T^{6} + 18 ⁣⋯ ⁣8418\!\cdots\!84T8− T^{8} - 93 ⁣⋯ ⁣2893\!\cdots\!28T10+ T^{10} + 40 ⁣⋯ ⁣1640\!\cdots\!16T12− T^{12} - 15 ⁣⋯ ⁣1215\!\cdots\!12T14+ T^{14} + 53 ⁣⋯ ⁣3453\!\cdots\!34T16− T^{16} - 37 ⁣⋯ ⁣7237\!\cdots\!72p2T18+ p^{2} T^{18} + 53 ⁣⋯ ⁣3453\!\cdots\!34p6T20− p^{6} T^{20} - 15 ⁣⋯ ⁣1215\!\cdots\!12p12T22+ p^{12} T^{22} + 40 ⁣⋯ ⁣1640\!\cdots\!16p18T24− p^{18} T^{24} - 93 ⁣⋯ ⁣2893\!\cdots\!28p24T26+ p^{24} T^{26} + 18 ⁣⋯ ⁣8418\!\cdots\!84p30T28−2996289198536038944p36T30+3533629828065p42T32−2700338p48T34+p54T36 p^{30} T^{28} - 2996289198536038944 p^{36} T^{30} + 3533629828065 p^{42} T^{32} - 2700338 p^{48} T^{34} + p^{54} T^{36}
71 (1−36T+1841615T2−165671520T3+1824835269476T4−185529115441136T5+1209991330264289612T6− ( 1 - 36 T + 1841615 T^{2} - 165671520 T^{3} + 1824835269476 T^{4} - 185529115441136 T^{5} + 1209991330264289612 T^{6} - 12 ⁣⋯ ⁣6012\!\cdots\!60T7+ T^{7} + 58 ⁣⋯ ⁣4658\!\cdots\!46T8− T^{8} - 53 ⁣⋯ ⁣3653\!\cdots\!36T9+ T^{9} + 58 ⁣⋯ ⁣4658\!\cdots\!46p3T10− p^{3} T^{10} - 12 ⁣⋯ ⁣6012\!\cdots\!60p6T11+1209991330264289612p9T12−185529115441136p12T13+1824835269476p15T14−165671520p18T15+1841615p21T16−36p24T17+p27T18)2 p^{6} T^{11} + 1209991330264289612 p^{9} T^{12} - 185529115441136 p^{12} T^{13} + 1824835269476 p^{15} T^{14} - 165671520 p^{18} T^{15} + 1841615 p^{21} T^{16} - 36 p^{24} T^{17} + p^{27} T^{18} )^{2}
73 1−3778834T2+6933266462601T4−8143954384598370864T6+ 1 - 3778834 T^{2} + 6933266462601 T^{4} - 8143954384598370864 T^{6} + 68 ⁣⋯ ⁣0868\!\cdots\!08T8− T^{8} - 42 ⁣⋯ ⁣4842\!\cdots\!48T10+ T^{10} + 20 ⁣⋯ ⁣1620\!\cdots\!16T12− T^{12} - 83 ⁣⋯ ⁣2883\!\cdots\!28T14+ T^{14} + 39 ⁣⋯ ⁣0239\!\cdots\!02pT16− p T^{16} - 10 ⁣⋯ ⁣1210\!\cdots\!12T18+ T^{18} + 39 ⁣⋯ ⁣0239\!\cdots\!02p7T20− p^{7} T^{20} - 83 ⁣⋯ ⁣2883\!\cdots\!28p12T22+ p^{12} T^{22} + 20 ⁣⋯ ⁣1620\!\cdots\!16p18T24− p^{18} T^{24} - 42 ⁣⋯ ⁣4842\!\cdots\!48p24T26+ p^{24} T^{26} + 68 ⁣⋯ ⁣0868\!\cdots\!08p30T28−8143954384598370864p36T30+6933266462601p42T32−3778834p48T34+p54T36 p^{30} T^{28} - 8143954384598370864 p^{36} T^{30} + 6933266462601 p^{42} T^{32} - 3778834 p^{48} T^{34} + p^{54} T^{36}
79 (1+976T+1521351T2+945533696T3+1131624911108T4+585675447712192T5+718720167172114636T6+ ( 1 + 976 T + 1521351 T^{2} + 945533696 T^{3} + 1131624911108 T^{4} + 585675447712192 T^{5} + 718720167172114636 T^{6} + 38 ⁣⋯ ⁣0038\!\cdots\!00T7+ T^{7} + 42 ⁣⋯ ⁣8242\!\cdots\!82T8+ T^{8} + 20 ⁣⋯ ⁣3220\!\cdots\!32T9+ T^{9} + 42 ⁣⋯ ⁣8242\!\cdots\!82p3T10+ p^{3} T^{10} + 38 ⁣⋯ ⁣0038\!\cdots\!00p6T11+718720167172114636p9T12+585675447712192p12T13+1131624911108p15T14+945533696p18T15+1521351p21T16+976p24T17+p27T18)2 p^{6} T^{11} + 718720167172114636 p^{9} T^{12} + 585675447712192 p^{12} T^{13} + 1131624911108 p^{15} T^{14} + 945533696 p^{18} T^{15} + 1521351 p^{21} T^{16} + 976 p^{24} T^{17} + p^{27} T^{18} )^{2}
83 1−6128690T2+18682450854913T4−37665709554087804320T6+ 1 - 6128690 T^{2} + 18682450854913 T^{4} - 37665709554087804320 T^{6} + 56 ⁣⋯ ⁣8456\!\cdots\!84T8− T^{8} - 66 ⁣⋯ ⁣5666\!\cdots\!56T10+ T^{10} + 65 ⁣⋯ ⁣3265\!\cdots\!32T12− T^{12} - 53 ⁣⋯ ⁣7653\!\cdots\!76T14+ T^{14} + 38 ⁣⋯ ⁣3438\!\cdots\!34T16− T^{16} - 23 ⁣⋯ ⁣3623\!\cdots\!36T18+ T^{18} + 38 ⁣⋯ ⁣3438\!\cdots\!34p6T20− p^{6} T^{20} - 53 ⁣⋯ ⁣7653\!\cdots\!76p12T22+ p^{12} T^{22} + 65 ⁣⋯ ⁣3265\!\cdots\!32p18T24− p^{18} T^{24} - 66 ⁣⋯ ⁣5666\!\cdots\!56p24T26+ p^{24} T^{26} + 56 ⁣⋯ ⁣8456\!\cdots\!84p30T28−37665709554087804320p36T30+18682450854913p42T32−6128690p48T34+p54T36 p^{30} T^{28} - 37665709554087804320 p^{36} T^{30} + 18682450854913 p^{42} T^{32} - 6128690 p^{48} T^{34} + p^{54} T^{36}
89 (1+146T+2947297T2+762766896T3+3967842683252T4+1157135497541624T5+3380357191051689636T6+ ( 1 + 146 T + 2947297 T^{2} + 762766896 T^{3} + 3967842683252 T^{4} + 1157135497541624 T^{5} + 3380357191051689636 T^{6} + 94 ⁣⋯ ⁣0094\!\cdots\!00T7+ T^{7} + 22 ⁣⋯ ⁣9022\!\cdots\!90T8+ T^{8} + 65 ⁣⋯ ⁣5265\!\cdots\!52T9+ T^{9} + 22 ⁣⋯ ⁣9022\!\cdots\!90p3T10+ p^{3} T^{10} + 94 ⁣⋯ ⁣0094\!\cdots\!00p6T11+3380357191051689636p9T12+1157135497541624p12T13+3967842683252p15T14+762766896p18T15+2947297p21T16+146p24T17+p27T18)2 p^{6} T^{11} + 3380357191051689636 p^{9} T^{12} + 1157135497541624 p^{12} T^{13} + 3967842683252 p^{15} T^{14} + 762766896 p^{18} T^{15} + 2947297 p^{21} T^{16} + 146 p^{24} T^{17} + p^{27} T^{18} )^{2}
97 1−10032834T2+49189296349913T4− 1 - 10032834 T^{2} + 49189296349913 T^{4} - 15 ⁣⋯ ⁣6015\!\cdots\!60T6+ T^{6} + 37 ⁣⋯ ⁣4437\!\cdots\!44T8− T^{8} - 70 ⁣⋯ ⁣4070\!\cdots\!40T10+ T^{10} + 11 ⁣⋯ ⁣6811\!\cdots\!68T12− T^{12} - 14 ⁣⋯ ⁣1614\!\cdots\!16T14+ T^{14} + 16 ⁣⋯ ⁣6616\!\cdots\!66T16− T^{16} - 16 ⁣⋯ ⁣2016\!\cdots\!20T18+ T^{18} + 16 ⁣⋯ ⁣6616\!\cdots\!66p6T20− p^{6} T^{20} - 14 ⁣⋯ ⁣1614\!\cdots\!16p12T22+ p^{12} T^{22} + 11 ⁣⋯ ⁣6811\!\cdots\!68p18T24− p^{18} T^{24} - 70 ⁣⋯ ⁣4070\!\cdots\!40p24T26+ p^{24} T^{26} + 37 ⁣⋯ ⁣4437\!\cdots\!44p30T28− p^{30} T^{28} - 15 ⁣⋯ ⁣6015\!\cdots\!60p36T30+49189296349913p42T32−10032834p48T34+p54T36 p^{36} T^{30} + 49189296349913 p^{42} T^{32} - 10032834 p^{48} T^{34} + p^{54} T^{36}
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   L(s)=∏p ∏j=136(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.04287912447139866452109918824, −2.03100533737620034523866477034, −1.99956505739540436389608576615, −1.81827007740901905616334864977, −1.80057522012279434504127529012, −1.78986357142706454380351000185, −1.68337665011858957549486716786, −1.64107525756995137527010571881, −1.53383533941417040400256407281, −1.26017319592086144907705238861, −1.19996960607242203654792337848, −1.12495821068424846665622980358, −1.11742235247817751858929908923, −1.08181103263074801848321188096, −1.03871456211604109219934306147, −0.975184275689600752277835990328, −0.943445877885009947284375611725, −0.925763777207148723006615527972, −0.65147725929841221453964575064, −0.58321847852207006038296654096, −0.45502747907187983524832467085, −0.31813568550163827952201827658, −0.26011434798141376845353399424, −0.04386617744531522675128674035, −0.01662599838095411594775182872, 0.01662599838095411594775182872, 0.04386617744531522675128674035, 0.26011434798141376845353399424, 0.31813568550163827952201827658, 0.45502747907187983524832467085, 0.58321847852207006038296654096, 0.65147725929841221453964575064, 0.925763777207148723006615527972, 0.943445877885009947284375611725, 0.975184275689600752277835990328, 1.03871456211604109219934306147, 1.08181103263074801848321188096, 1.11742235247817751858929908923, 1.12495821068424846665622980358, 1.19996960607242203654792337848, 1.26017319592086144907705238861, 1.53383533941417040400256407281, 1.64107525756995137527010571881, 1.68337665011858957549486716786, 1.78986357142706454380351000185, 1.80057522012279434504127529012, 1.81827007740901905616334864977, 1.99956505739540436389608576615, 2.03100533737620034523866477034, 2.04287912447139866452109918824

Graph of the ZZ-function along the critical line

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