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Properties

Label 24-1183e12-1.1-c1e12-0-3
Degree 2424
Conductor 7.513×10367.513\times 10^{36}
Sign 11
Analytic cond. 5.04826×10115.04826\times 10^{11}
Root an. cond. 3.073483.07348
Motivic weight 11
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  + 2·2-s + 3-s + 6·4-s + 5-s + 2·6-s − 6·7-s + 10·8-s + 11·9-s + 2·10-s + 4·11-s + 6·12-s − 12·14-s + 15-s + 22·16-s + 5·17-s + 22·18-s − 19-s + 6·20-s − 6·21-s + 8·22-s − 23-s + 10·24-s + 19·25-s + 10·27-s − 36·28-s − 6·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3·4-s + 0.447·5-s + 0.816·6-s − 2.26·7-s + 3.53·8-s + 11/3·9-s + 0.632·10-s + 1.20·11-s + 1.73·12-s − 3.20·14-s + 0.258·15-s + 11/2·16-s + 1.21·17-s + 5.18·18-s − 0.229·19-s + 1.34·20-s − 1.30·21-s + 1.70·22-s − 0.208·23-s + 2.04·24-s + 19/5·25-s + 1.92·27-s − 6.80·28-s − 1.11·29-s + 0.365·30-s + ⋯

Functional equation

Λ(s)=((712⋅1324)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((712⋅1324)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 712⋅13247^{12} \cdot 13^{24}
Sign: 11
Analytic conductor: 5.04826×10115.04826\times 10^{11}
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 712⋅1324, ( :[1/2]12), 1)(24,\ 7^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 516.7067971516.7067971
L(12)L(\frac12) ≈\approx 516.7067971516.7067971
L(32)L(\frac{3}{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)
bad7 1+6T+15T2+54T3+219T4+516T5+1069T6+516pT7+219p2T8+54p3T9+15p4T10+6p5T11+p6T12 1 + 6 T + 15 T^{2} + 54 T^{3} + 219 T^{4} + 516 T^{5} + 1069 T^{6} + 516 p T^{7} + 219 p^{2} T^{8} + 54 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
13 1 1
good2 1−pT−pT2+3pT3−pT4−pT5+7T6−pT7−7pT8+3pT9−5p2T10−5p2T11+153T12−5p3T13−5p4T14+3p4T15−7p5T16−p6T17+7p6T18−p8T19−p9T20+3p10T21−p11T22−p12T23+p12T24 1 - p T - p T^{2} + 3 p T^{3} - p T^{4} - p T^{5} + 7 T^{6} - p T^{7} - 7 p T^{8} + 3 p T^{9} - 5 p^{2} T^{10} - 5 p^{2} T^{11} + 153 T^{12} - 5 p^{3} T^{13} - 5 p^{4} T^{14} + 3 p^{4} T^{15} - 7 p^{5} T^{16} - p^{6} T^{17} + 7 p^{6} T^{18} - p^{8} T^{19} - p^{9} T^{20} + 3 p^{10} T^{21} - p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24}
3 1−T−10T2+11T3+53T4−62T5−167T6+221T7+98pT8−535T9+79T10+604T11−1559T12+604pT13+79p2T14−535p3T15+98p5T16+221p5T17−167p6T18−62p7T19+53p8T20+11p9T21−10p10T22−p11T23+p12T24 1 - T - 10 T^{2} + 11 T^{3} + 53 T^{4} - 62 T^{5} - 167 T^{6} + 221 T^{7} + 98 p T^{8} - 535 T^{9} + 79 T^{10} + 604 T^{11} - 1559 T^{12} + 604 p T^{13} + 79 p^{2} T^{14} - 535 p^{3} T^{15} + 98 p^{5} T^{16} + 221 p^{5} T^{17} - 167 p^{6} T^{18} - 62 p^{7} T^{19} + 53 p^{8} T^{20} + 11 p^{9} T^{21} - 10 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24}
5 1−T−18T2−pT3+193T4+192T5−1181T6−2139T7+908pT8+451p2T9−6679T10−26266T11+249T12−26266pT13−6679p2T14+451p5T15+908p5T16−2139p5T17−1181p6T18+192p7T19+193p8T20−p10T21−18p10T22−p11T23+p12T24 1 - T - 18 T^{2} - p T^{3} + 193 T^{4} + 192 T^{5} - 1181 T^{6} - 2139 T^{7} + 908 p T^{8} + 451 p^{2} T^{9} - 6679 T^{10} - 26266 T^{11} + 249 T^{12} - 26266 p T^{13} - 6679 p^{2} T^{14} + 451 p^{5} T^{15} + 908 p^{5} T^{16} - 2139 p^{5} T^{17} - 1181 p^{6} T^{18} + 192 p^{7} T^{19} + 193 p^{8} T^{20} - p^{10} T^{21} - 18 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24}
11 1−4T−29T2+108T3+477T4−113pT5−6686T6+7665T7+89323T8+423T9−1282040T10−249219T11+16505087T12−249219pT13−1282040p2T14+423p3T15+89323p4T16+7665p5T17−6686p6T18−113p8T19+477p8T20+108p9T21−29p10T22−4p11T23+p12T24 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24}
17 1−5T−65T2+372T3+2506T4−15344T5−62063T6+395128T7+1158376T8−6526599T9−17123414T10+46016896T11+268434807T12+46016896pT13−17123414p2T14−6526599p3T15+1158376p4T16+395128p5T17−62063p6T18−15344p7T19+2506p8T20+372p9T21−65p10T22−5p11T23+p12T24 1 - 5 T - 65 T^{2} + 372 T^{3} + 2506 T^{4} - 15344 T^{5} - 62063 T^{6} + 395128 T^{7} + 1158376 T^{8} - 6526599 T^{9} - 17123414 T^{10} + 46016896 T^{11} + 268434807 T^{12} + 46016896 p T^{13} - 17123414 p^{2} T^{14} - 6526599 p^{3} T^{15} + 1158376 p^{4} T^{16} + 395128 p^{5} T^{17} - 62063 p^{6} T^{18} - 15344 p^{7} T^{19} + 2506 p^{8} T^{20} + 372 p^{9} T^{21} - 65 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24}
19 1+T−49T2+82T3+1336T4−4335T5−10907T6+99626T7−263580T8−1110690T9+9684539T10+2414194T11−215227743T12+2414194pT13+9684539p2T14−1110690p3T15−263580p4T16+99626p5T17−10907p6T18−4335p7T19+1336p8T20+82p9T21−49p10T22+p11T23+p12T24 1 + T - 49 T^{2} + 82 T^{3} + 1336 T^{4} - 4335 T^{5} - 10907 T^{6} + 99626 T^{7} - 263580 T^{8} - 1110690 T^{9} + 9684539 T^{10} + 2414194 T^{11} - 215227743 T^{12} + 2414194 p T^{13} + 9684539 p^{2} T^{14} - 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} + 99626 p^{5} T^{17} - 10907 p^{6} T^{18} - 4335 p^{7} T^{19} + 1336 p^{8} T^{20} + 82 p^{9} T^{21} - 49 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
23 1+T−31T2−72T3−242T4+619T5+6343T6+28360T7+121372T8−9024pT9+5143119T10−4729028T11−273608319T12−4729028pT13+5143119p2T14−9024p4T15+121372p4T16+28360p5T17+6343p6T18+619p7T19−242p8T20−72p9T21−31p10T22+p11T23+p12T24 1 + T - 31 T^{2} - 72 T^{3} - 242 T^{4} + 619 T^{5} + 6343 T^{6} + 28360 T^{7} + 121372 T^{8} - 9024 p T^{9} + 5143119 T^{10} - 4729028 T^{11} - 273608319 T^{12} - 4729028 p T^{13} + 5143119 p^{2} T^{14} - 9024 p^{4} T^{15} + 121372 p^{4} T^{16} + 28360 p^{5} T^{17} + 6343 p^{6} T^{18} + 619 p^{7} T^{19} - 242 p^{8} T^{20} - 72 p^{9} T^{21} - 31 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
29 (1+3T+96T2+191T3+4061T4+5126T5+122643T6+5126pT7+4061p2T8+191p3T9+96p4T10+3p5T11+p6T12)2 ( 1 + 3 T + 96 T^{2} + 191 T^{3} + 4061 T^{4} + 5126 T^{5} + 122643 T^{6} + 5126 p T^{7} + 4061 p^{2} T^{8} + 191 p^{3} T^{9} + 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
31 1−16T+20T2+594T3+2163T4−43649T5−56125T6+1282696T7+2984747T8−22743273T9−180497697T10+655302586T11+2182678017T12+655302586pT13−180497697p2T14−22743273p3T15+2984747p4T16+1282696p5T17−56125p6T18−43649p7T19+2163p8T20+594p9T21+20p10T22−16p11T23+p12T24 1 - 16 T + 20 T^{2} + 594 T^{3} + 2163 T^{4} - 43649 T^{5} - 56125 T^{6} + 1282696 T^{7} + 2984747 T^{8} - 22743273 T^{9} - 180497697 T^{10} + 655302586 T^{11} + 2182678017 T^{12} + 655302586 p T^{13} - 180497697 p^{2} T^{14} - 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} + 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} - 43649 p^{7} T^{19} + 2163 p^{8} T^{20} + 594 p^{9} T^{21} + 20 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24}
37 1+13T−15T2+284T3+12996T4+18401T5−116147T6+5523346T7+19538810T8−71463812T9+1452640399T10+7689412934T11−18842100883T12+7689412934pT13+1452640399p2T14−71463812p3T15+19538810p4T16+5523346p5T17−116147p6T18+18401p7T19+12996p8T20+284p9T21−15p10T22+13p11T23+p12T24 1 + 13 T - 15 T^{2} + 284 T^{3} + 12996 T^{4} + 18401 T^{5} - 116147 T^{6} + 5523346 T^{7} + 19538810 T^{8} - 71463812 T^{9} + 1452640399 T^{10} + 7689412934 T^{11} - 18842100883 T^{12} + 7689412934 p T^{13} + 1452640399 p^{2} T^{14} - 71463812 p^{3} T^{15} + 19538810 p^{4} T^{16} + 5523346 p^{5} T^{17} - 116147 p^{6} T^{18} + 18401 p^{7} T^{19} + 12996 p^{8} T^{20} + 284 p^{9} T^{21} - 15 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24}
41 (1−8T+225T2−1362T3+21488T4−101725T5+1145451T6−101725pT7+21488p2T8−1362p3T9+225p4T10−8p5T11+p6T12)2 ( 1 - 8 T + 225 T^{2} - 1362 T^{3} + 21488 T^{4} - 101725 T^{5} + 1145451 T^{6} - 101725 p T^{7} + 21488 p^{2} T^{8} - 1362 p^{3} T^{9} + 225 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2}
43 (1−11T+259T2−2099T3+27622T4−170696T5+1576761T6−170696pT7+27622p2T8−2099p3T9+259p4T10−11p5T11+p6T12)2 ( 1 - 11 T + 259 T^{2} - 2099 T^{3} + 27622 T^{4} - 170696 T^{5} + 1576761 T^{6} - 170696 p T^{7} + 27622 p^{2} T^{8} - 2099 p^{3} T^{9} + 259 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2}
47 1+T−104T2−189T3+5335T4+164pT5−69863T6+514255T7−7627520T8−55687467T9+662939941T10+1686387922T11−35399065407T12+1686387922pT13+662939941p2T14−55687467p3T15−7627520p4T16+514255p5T17−69863p6T18+164p8T19+5335p8T20−189p9T21−104p10T22+p11T23+p12T24 1 + T - 104 T^{2} - 189 T^{3} + 5335 T^{4} + 164 p T^{5} - 69863 T^{6} + 514255 T^{7} - 7627520 T^{8} - 55687467 T^{9} + 662939941 T^{10} + 1686387922 T^{11} - 35399065407 T^{12} + 1686387922 p T^{13} + 662939941 p^{2} T^{14} - 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} + 514255 p^{5} T^{17} - 69863 p^{6} T^{18} + 164 p^{8} T^{19} + 5335 p^{8} T^{20} - 189 p^{9} T^{21} - 104 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24}
53 1+2T−214T2−252T3+24796T4+13772T5−1921862T6+82142T7+113089342T8−43114584T9−5653831794T10+1443208718T11+285781391787T12+1443208718pT13−5653831794p2T14−43114584p3T15+113089342p4T16+82142p5T17−1921862p6T18+13772p7T19+24796p8T20−252p9T21−214p10T22+2p11T23+p12T24 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
59 1−13T−126T2+1843T3+11161T4−119322T5−1337447T6+7367025T7+123366322T8−382593671T9−9090177085T10+8156701016T11+592237594305T12+8156701016pT13−9090177085p2T14−382593671p3T15+123366322p4T16+7367025p5T17−1337447p6T18−119322p7T19+11161p8T20+1843p9T21−126p10T22−13p11T23+p12T24 1 - 13 T - 126 T^{2} + 1843 T^{3} + 11161 T^{4} - 119322 T^{5} - 1337447 T^{6} + 7367025 T^{7} + 123366322 T^{8} - 382593671 T^{9} - 9090177085 T^{10} + 8156701016 T^{11} + 592237594305 T^{12} + 8156701016 p T^{13} - 9090177085 p^{2} T^{14} - 382593671 p^{3} T^{15} + 123366322 p^{4} T^{16} + 7367025 p^{5} T^{17} - 1337447 p^{6} T^{18} - 119322 p^{7} T^{19} + 11161 p^{8} T^{20} + 1843 p^{9} T^{21} - 126 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24}
61 1+5T−140T2−373T3+8487T4−5202T5−147441T6+963135T7−4711566T8−13690661T9−1296684385T10−689962304T11+162150963097T12−689962304pT13−1296684385p2T14−13690661p3T15−4711566p4T16+963135p5T17−147441p6T18−5202p7T19+8487p8T20−373p9T21−140p10T22+5p11T23+p12T24 1 + 5 T - 140 T^{2} - 373 T^{3} + 8487 T^{4} - 5202 T^{5} - 147441 T^{6} + 963135 T^{7} - 4711566 T^{8} - 13690661 T^{9} - 1296684385 T^{10} - 689962304 T^{11} + 162150963097 T^{12} - 689962304 p T^{13} - 1296684385 p^{2} T^{14} - 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} + 963135 p^{5} T^{17} - 147441 p^{6} T^{18} - 5202 p^{7} T^{19} + 8487 p^{8} T^{20} - 373 p^{9} T^{21} - 140 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24}
67 1+11T−175T2−2336T3+15663T4+247450T5−15954pT6−18125445T7+60512732T8+977936543T9−2490157221T10−26393757979T11+95373451231T12−26393757979pT13−2490157221p2T14+977936543p3T15+60512732p4T16−18125445p5T17−15954p7T18+247450p7T19+15663p8T20−2336p9T21−175p10T22+11p11T23+p12T24 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24}
71 (1+6T+285T2+14pT3+35468T4+74185T5+2901951T6+74185pT7+35468p2T8+14p4T9+285p4T10+6p5T11+p6T12)2 ( 1 + 6 T + 285 T^{2} + 14 p T^{3} + 35468 T^{4} + 74185 T^{5} + 2901951 T^{6} + 74185 p T^{7} + 35468 p^{2} T^{8} + 14 p^{4} T^{9} + 285 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2}
73 1+30T+224T2−1118T3−5021T4+290169T5+1854677T6−9817892T7−27971653T8+688598777T9−1819010273T10−20701972840T11+235631264151T12−20701972840pT13−1819010273p2T14+688598777p3T15−27971653p4T16−9817892p5T17+1854677p6T18+290169p7T19−5021p8T20−1118p9T21+224p10T22+30p11T23+p12T24 1 + 30 T + 224 T^{2} - 1118 T^{3} - 5021 T^{4} + 290169 T^{5} + 1854677 T^{6} - 9817892 T^{7} - 27971653 T^{8} + 688598777 T^{9} - 1819010273 T^{10} - 20701972840 T^{11} + 235631264151 T^{12} - 20701972840 p T^{13} - 1819010273 p^{2} T^{14} + 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} - 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} + 290169 p^{7} T^{19} - 5021 p^{8} T^{20} - 1118 p^{9} T^{21} + 224 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24}
79 1−7T−277T2+2628T3+34995T4−387429T5−3070086T6+26237658T7+339376855T8−565746882T9−45365142063T10−8895648284T11+4474615429807T12−8895648284pT13−45365142063p2T14−565746882p3T15+339376855p4T16+26237658p5T17−3070086p6T18−387429p7T19+34995p8T20+2628p9T21−277p10T22−7p11T23+p12T24 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24}
83 (1+27T+656T2+10802T3+153994T4+1760871T5+17670883T6+1760871pT7+153994p2T8+10802p3T9+656p4T10+27p5T11+p6T12)2 ( 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2}
89 1−4T−151T2+2628T3+262T4−309046T5+2802769T6+6034970T7−281402495T8+1666391304T9+5367237150T10−95837476354T11+701675320941T12−95837476354pT13+5367237150p2T14+1666391304p3T15−281402495p4T16+6034970p5T17+2802769p6T18−309046p7T19+262p8T20+2628p9T21−151p10T22−4p11T23+p12T24 1 - 4 T - 151 T^{2} + 2628 T^{3} + 262 T^{4} - 309046 T^{5} + 2802769 T^{6} + 6034970 T^{7} - 281402495 T^{8} + 1666391304 T^{9} + 5367237150 T^{10} - 95837476354 T^{11} + 701675320941 T^{12} - 95837476354 p T^{13} + 5367237150 p^{2} T^{14} + 1666391304 p^{3} T^{15} - 281402495 p^{4} T^{16} + 6034970 p^{5} T^{17} + 2802769 p^{6} T^{18} - 309046 p^{7} T^{19} + 262 p^{8} T^{20} + 2628 p^{9} T^{21} - 151 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24}
97 (1−35T+947T2−18161T3+281670T4−3629766T5+38644781T6−3629766pT7+281670p2T8−18161p3T9+947p4T10−35p5T11+p6T12)2 ( 1 - 35 T + 947 T^{2} - 18161 T^{3} + 281670 T^{4} - 3629766 T^{5} + 38644781 T^{6} - 3629766 p T^{7} + 281670 p^{2} T^{8} - 18161 p^{3} T^{9} + 947 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} )^{2}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.13679713754667639444242213842, −3.04542252205656983029221781959, −2.85949635619482549788466658570, −2.85446628604575294612051890057, −2.78283243420235661771960741806, −2.65223387478996395545338192992, −2.61421934187211140168286093510, −2.58637762954337047507637185959, −2.39674073322770788286633990806, −2.18393441743156440932815716528, −1.98717490099525154133052795015, −1.87410683825105462642809504180, −1.86498835157656590535755585547, −1.79404808333237189492533285548, −1.77495561398805324496363258538, −1.76556814732911963227987015914, −1.62481084358282326141045090172, −1.29710065117532689406433266524, −1.16527599217888866878367137103, −0.882986977365847155704218535426, −0.875041436693927283282975098719, −0.826931056324815880384631235460, −0.822757818520953141613265308000, −0.67078506398929520254109184985, −0.46661322384760203045745736940, 0.46661322384760203045745736940, 0.67078506398929520254109184985, 0.822757818520953141613265308000, 0.826931056324815880384631235460, 0.875041436693927283282975098719, 0.882986977365847155704218535426, 1.16527599217888866878367137103, 1.29710065117532689406433266524, 1.62481084358282326141045090172, 1.76556814732911963227987015914, 1.77495561398805324496363258538, 1.79404808333237189492533285548, 1.86498835157656590535755585547, 1.87410683825105462642809504180, 1.98717490099525154133052795015, 2.18393441743156440932815716528, 2.39674073322770788286633990806, 2.58637762954337047507637185959, 2.61421934187211140168286093510, 2.65223387478996395545338192992, 2.78283243420235661771960741806, 2.85446628604575294612051890057, 2.85949635619482549788466658570, 3.04542252205656983029221781959, 3.13679713754667639444242213842

Graph of the ZZ-function along the critical line

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