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Properties

Label 24-3e72-1.1-c1e12-0-0
Degree 2424
Conductor 2.253×10342.253\times 10^{34}
Sign 11
Analytic cond. 1.51375×1091.51375\times 10^{9}
Root an. cond. 2.412692.41269
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  + 3·2-s + 6·4-s − 3·5-s − 6·7-s + 9·8-s − 9·10-s − 6·11-s − 6·13-s − 18·14-s + 12·16-s + 18·17-s + 24·19-s − 18·20-s − 18·22-s − 12·23-s + 15·25-s − 18·26-s − 36·28-s + 21·29-s − 15·31-s + 12·32-s + 54·34-s + 18·35-s + 6·37-s + 72·38-s − 27·40-s − 12·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s − 2.26·7-s + 3.18·8-s − 2.84·10-s − 1.80·11-s − 1.66·13-s − 4.81·14-s + 3·16-s + 4.36·17-s + 5.50·19-s − 4.02·20-s − 3.83·22-s − 2.50·23-s + 3·25-s − 3.53·26-s − 6.80·28-s + 3.89·29-s − 2.69·31-s + 2.12·32-s + 9.26·34-s + 3.04·35-s + 0.986·37-s + 11.6·38-s − 4.26·40-s − 1.87·41-s + ⋯

Functional equation

Λ(s)=((372)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((372)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 3723^{72}
Sign: 11
Analytic conductor: 1.51375×1091.51375\times 10^{9}
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 372, ( :[1/2]12), 1)(24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.52553325150.5255332515
L(12)L(\frac12) ≈\approx 0.52553325150.5255332515
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)
bad3 1 1
good2 1−3T+3T2−3T4+3pT5−T6−27T7+27pT8−27pT9+9p2T10+9p2T11−135T12+9p3T13+9p4T14−27p4T15+27p5T16−27p5T17−p6T18+3p8T19−3p8T20+3p10T22−3p11T23+p12T24 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - T^{6} - 27 T^{7} + 27 p T^{8} - 27 p T^{9} + 9 p^{2} T^{10} + 9 p^{2} T^{11} - 135 T^{12} + 9 p^{3} T^{13} + 9 p^{4} T^{14} - 27 p^{4} T^{15} + 27 p^{5} T^{16} - 27 p^{5} T^{17} - p^{6} T^{18} + 3 p^{8} T^{19} - 3 p^{8} T^{20} + 3 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24}
5 1+3T−6T2+9T3+87T4−6p2T5−38pT6+909T7−1521T8−2808T9+891pT10−3969T11−19179T12−3969pT13+891p3T14−2808p3T15−1521p4T16+909p5T17−38p7T18−6p9T19+87p8T20+9p9T21−6p10T22+3p11T23+p12T24 1 + 3 T - 6 T^{2} + 9 T^{3} + 87 T^{4} - 6 p^{2} T^{5} - 38 p T^{6} + 909 T^{7} - 1521 T^{8} - 2808 T^{9} + 891 p T^{10} - 3969 T^{11} - 19179 T^{12} - 3969 p T^{13} + 891 p^{3} T^{14} - 2808 p^{3} T^{15} - 1521 p^{4} T^{16} + 909 p^{5} T^{17} - 38 p^{7} T^{18} - 6 p^{9} T^{19} + 87 p^{8} T^{20} + 9 p^{9} T^{21} - 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24}
7 1+6T+3T2−8pT3−150T4−48T5+10T6−1440T7−477T8+20932T9+52197T10−25890T11−306689T12−25890pT13+52197p2T14+20932p3T15−477p4T16−1440p5T17+10p6T18−48p7T19−150p8T20−8p10T21+3p10T22+6p11T23+p12T24 1 + 6 T + 3 T^{2} - 8 p T^{3} - 150 T^{4} - 48 T^{5} + 10 T^{6} - 1440 T^{7} - 477 T^{8} + 20932 T^{9} + 52197 T^{10} - 25890 T^{11} - 306689 T^{12} - 25890 p T^{13} + 52197 p^{2} T^{14} + 20932 p^{3} T^{15} - 477 p^{4} T^{16} - 1440 p^{5} T^{17} + 10 p^{6} T^{18} - 48 p^{7} T^{19} - 150 p^{8} T^{20} - 8 p^{10} T^{21} + 3 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
11 1+6T−15T2−72T3+402T4+114T5−8848T6−7092T7+73287T8+29538T9−873405T10+573048T11+15237891T12+573048pT13−873405p2T14+29538p3T15+73287p4T16−7092p5T17−8848p6T18+114p7T19+402p8T20−72p9T21−15p10T22+6p11T23+p12T24 1 + 6 T - 15 T^{2} - 72 T^{3} + 402 T^{4} + 114 T^{5} - 8848 T^{6} - 7092 T^{7} + 73287 T^{8} + 29538 T^{9} - 873405 T^{10} + 573048 T^{11} + 15237891 T^{12} + 573048 p T^{13} - 873405 p^{2} T^{14} + 29538 p^{3} T^{15} + 73287 p^{4} T^{16} - 7092 p^{5} T^{17} - 8848 p^{6} T^{18} + 114 p^{7} T^{19} + 402 p^{8} T^{20} - 72 p^{9} T^{21} - 15 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
13 1+6T−24T2−272T3+30T4+6234T5+10342T6−93456T7−311490T8+964312T9+5908470T10−4890696T11−86990081T12−4890696pT13+5908470p2T14+964312p3T15−311490p4T16−93456p5T17+10342p6T18+6234p7T19+30p8T20−272p9T21−24p10T22+6p11T23+p12T24 1 + 6 T - 24 T^{2} - 272 T^{3} + 30 T^{4} + 6234 T^{5} + 10342 T^{6} - 93456 T^{7} - 311490 T^{8} + 964312 T^{9} + 5908470 T^{10} - 4890696 T^{11} - 86990081 T^{12} - 4890696 p T^{13} + 5908470 p^{2} T^{14} + 964312 p^{3} T^{15} - 311490 p^{4} T^{16} - 93456 p^{5} T^{17} + 10342 p^{6} T^{18} + 6234 p^{7} T^{19} + 30 p^{8} T^{20} - 272 p^{9} T^{21} - 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
17 (1−9T+6pT2−630T3+4227T4−19611T5+5591pT6−19611pT7+4227p2T8−630p3T9+6p5T10−9p5T11+p6T12)2 ( 1 - 9 T + 6 p T^{2} - 630 T^{3} + 4227 T^{4} - 19611 T^{5} + 5591 p T^{6} - 19611 p T^{7} + 4227 p^{2} T^{8} - 630 p^{3} T^{9} + 6 p^{5} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 (1−12T+129T2−985T3+6471T4−35073T5+166182T6−35073pT7+6471p2T8−985p3T9+129p4T10−12p5T11+p6T12)2 ( 1 - 12 T + 129 T^{2} - 985 T^{3} + 6471 T^{4} - 35073 T^{5} + 166182 T^{6} - 35073 p T^{7} + 6471 p^{2} T^{8} - 985 p^{3} T^{9} + 129 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2}
23 1+12T−15T2−450T3+2022T4+18012T5−97840T6−468468T7+3500631T8+7484670T9−108075537T10−86356332T11+2494857051T12−86356332pT13−108075537p2T14+7484670p3T15+3500631p4T16−468468p5T17−97840p6T18+18012p7T19+2022p8T20−450p9T21−15p10T22+12p11T23+p12T24 1 + 12 T - 15 T^{2} - 450 T^{3} + 2022 T^{4} + 18012 T^{5} - 97840 T^{6} - 468468 T^{7} + 3500631 T^{8} + 7484670 T^{9} - 108075537 T^{10} - 86356332 T^{11} + 2494857051 T^{12} - 86356332 p T^{13} - 108075537 p^{2} T^{14} + 7484670 p^{3} T^{15} + 3500631 p^{4} T^{16} - 468468 p^{5} T^{17} - 97840 p^{6} T^{18} + 18012 p^{7} T^{19} + 2022 p^{8} T^{20} - 450 p^{9} T^{21} - 15 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24}
29 1−21T+156T2−333T3−1335T4+11436T5−99298T6+512379T7+1250271T8−18237096T9+38476611T10−247254327T11+2704348845T12−247254327pT13+38476611p2T14−18237096p3T15+1250271p4T16+512379p5T17−99298p6T18+11436p7T19−1335p8T20−333p9T21+156p10T22−21p11T23+p12T24 1 - 21 T + 156 T^{2} - 333 T^{3} - 1335 T^{4} + 11436 T^{5} - 99298 T^{6} + 512379 T^{7} + 1250271 T^{8} - 18237096 T^{9} + 38476611 T^{10} - 247254327 T^{11} + 2704348845 T^{12} - 247254327 p T^{13} + 38476611 p^{2} T^{14} - 18237096 p^{3} T^{15} + 1250271 p^{4} T^{16} + 512379 p^{5} T^{17} - 99298 p^{6} T^{18} + 11436 p^{7} T^{19} - 1335 p^{8} T^{20} - 333 p^{9} T^{21} + 156 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24}
31 1+15T−6T2−731T3+3234T4+45879T5−179792T6−1417725T7+10709046T8+30200209T9−495230610T10−533330277T11+15229407838T12−533330277pT13−495230610p2T14+30200209p3T15+10709046p4T16−1417725p5T17−179792p6T18+45879p7T19+3234p8T20−731p9T21−6p10T22+15p11T23+p12T24 1 + 15 T - 6 T^{2} - 731 T^{3} + 3234 T^{4} + 45879 T^{5} - 179792 T^{6} - 1417725 T^{7} + 10709046 T^{8} + 30200209 T^{9} - 495230610 T^{10} - 533330277 T^{11} + 15229407838 T^{12} - 533330277 p T^{13} - 495230610 p^{2} T^{14} + 30200209 p^{3} T^{15} + 10709046 p^{4} T^{16} - 1417725 p^{5} T^{17} - 179792 p^{6} T^{18} + 45879 p^{7} T^{19} + 3234 p^{8} T^{20} - 731 p^{9} T^{21} - 6 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24}
37 (1−3T+93T2−94T3+4608T4+864T5+171555T6+864pT7+4608p2T8−94p3T9+93p4T10−3p5T11+p6T12)2 ( 1 - 3 T + 93 T^{2} - 94 T^{3} + 4608 T^{4} + 864 T^{5} + 171555 T^{6} + 864 p T^{7} + 4608 p^{2} T^{8} - 94 p^{3} T^{9} + 93 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
41 1+12T−42T2−990T3+15T4+44292T5+3617T6−2617299T7−8664444T8+83406402T9+621970299T10−647096049T11−21957305292T12−647096049pT13+621970299p2T14+83406402p3T15−8664444p4T16−2617299p5T17+3617p6T18+44292p7T19+15p8T20−990p9T21−42p10T22+12p11T23+p12T24 1 + 12 T - 42 T^{2} - 990 T^{3} + 15 T^{4} + 44292 T^{5} + 3617 T^{6} - 2617299 T^{7} - 8664444 T^{8} + 83406402 T^{9} + 621970299 T^{10} - 647096049 T^{11} - 21957305292 T^{12} - 647096049 p T^{13} + 621970299 p^{2} T^{14} + 83406402 p^{3} T^{15} - 8664444 p^{4} T^{16} - 2617299 p^{5} T^{17} + 3617 p^{6} T^{18} + 44292 p^{7} T^{19} + 15 p^{8} T^{20} - 990 p^{9} T^{21} - 42 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24}
43 1+6T−114T2−416T3+7617T4+10428T5−340838T6−469998T7+8946594T8+23651170T9−277922874T10−369502188T11+14787611473T12−369502188pT13−277922874p2T14+23651170p3T15+8946594p4T16−469998p5T17−340838p6T18+10428p7T19+7617p8T20−416p9T21−114p10T22+6p11T23+p12T24 1 + 6 T - 114 T^{2} - 416 T^{3} + 7617 T^{4} + 10428 T^{5} - 340838 T^{6} - 469998 T^{7} + 8946594 T^{8} + 23651170 T^{9} - 277922874 T^{10} - 369502188 T^{11} + 14787611473 T^{12} - 369502188 p T^{13} - 277922874 p^{2} T^{14} + 23651170 p^{3} T^{15} + 8946594 p^{4} T^{16} - 469998 p^{5} T^{17} - 340838 p^{6} T^{18} + 10428 p^{7} T^{19} + 7617 p^{8} T^{20} - 416 p^{9} T^{21} - 114 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
47 1+15T−15T2−1332T3−4053T4+50577T5+149399T6−2935773T7−11979270T8+128353842T9+985854933T10−1739075679T11−45888436269T12−1739075679pT13+985854933p2T14+128353842p3T15−11979270p4T16−2935773p5T17+149399p6T18+50577p7T19−4053p8T20−1332p9T21−15p10T22+15p11T23+p12T24 1 + 15 T - 15 T^{2} - 1332 T^{3} - 4053 T^{4} + 50577 T^{5} + 149399 T^{6} - 2935773 T^{7} - 11979270 T^{8} + 128353842 T^{9} + 985854933 T^{10} - 1739075679 T^{11} - 45888436269 T^{12} - 1739075679 p T^{13} + 985854933 p^{2} T^{14} + 128353842 p^{3} T^{15} - 11979270 p^{4} T^{16} - 2935773 p^{5} T^{17} + 149399 p^{6} T^{18} + 50577 p^{7} T^{19} - 4053 p^{8} T^{20} - 1332 p^{9} T^{21} - 15 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24}
53 (1−9T+237T2−1656T3+26583T4−151479T5+1786030T6−151479pT7+26583p2T8−1656p3T9+237p4T10−9p5T11+p6T12)2 ( 1 - 9 T + 237 T^{2} - 1656 T^{3} + 26583 T^{4} - 151479 T^{5} + 1786030 T^{6} - 151479 p T^{7} + 26583 p^{2} T^{8} - 1656 p^{3} T^{9} + 237 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
59 1−6T−213T2+1296T3+25512T4−153816T5−1929232T6+11972142T7+97225533T8−608531454T9−3222929547T10+14424181056T11+112233818043T12+14424181056pT13−3222929547p2T14−608531454p3T15+97225533p4T16+11972142p5T17−1929232p6T18−153816p7T19+25512p8T20+1296p9T21−213p10T22−6p11T23+p12T24 1 - 6 T - 213 T^{2} + 1296 T^{3} + 25512 T^{4} - 153816 T^{5} - 1929232 T^{6} + 11972142 T^{7} + 97225533 T^{8} - 608531454 T^{9} - 3222929547 T^{10} + 14424181056 T^{11} + 112233818043 T^{12} + 14424181056 p T^{13} - 3222929547 p^{2} T^{14} - 608531454 p^{3} T^{15} + 97225533 p^{4} T^{16} + 11972142 p^{5} T^{17} - 1929232 p^{6} T^{18} - 153816 p^{7} T^{19} + 25512 p^{8} T^{20} + 1296 p^{9} T^{21} - 213 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24}
61 1+24T+93T2−1946T3−10266T4+172446T5+1060300T6−9090504T7−44005779T8+700679680T9+78417087pT10−13840719090T11−283034464943T12−13840719090pT13+78417087p3T14+700679680p3T15−44005779p4T16−9090504p5T17+1060300p6T18+172446p7T19−10266p8T20−1946p9T21+93p10T22+24p11T23+p12T24 1 + 24 T + 93 T^{2} - 1946 T^{3} - 10266 T^{4} + 172446 T^{5} + 1060300 T^{6} - 9090504 T^{7} - 44005779 T^{8} + 700679680 T^{9} + 78417087 p T^{10} - 13840719090 T^{11} - 283034464943 T^{12} - 13840719090 p T^{13} + 78417087 p^{3} T^{14} + 700679680 p^{3} T^{15} - 44005779 p^{4} T^{16} - 9090504 p^{5} T^{17} + 1060300 p^{6} T^{18} + 172446 p^{7} T^{19} - 10266 p^{8} T^{20} - 1946 p^{9} T^{21} + 93 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24}
67 1+15T−123T2−2792T3+6357T4+236553T5−637361T6−187659pT7+114802740T8+538384030T9−12717543819T10−12718124403T11+988677433003T12−12718124403pT13−12717543819p2T14+538384030p3T15+114802740p4T16−187659p6T17−637361p6T18+236553p7T19+6357p8T20−2792p9T21−123p10T22+15p11T23+p12T24 1 + 15 T - 123 T^{2} - 2792 T^{3} + 6357 T^{4} + 236553 T^{5} - 637361 T^{6} - 187659 p T^{7} + 114802740 T^{8} + 538384030 T^{9} - 12717543819 T^{10} - 12718124403 T^{11} + 988677433003 T^{12} - 12718124403 p T^{13} - 12717543819 p^{2} T^{14} + 538384030 p^{3} T^{15} + 114802740 p^{4} T^{16} - 187659 p^{6} T^{17} - 637361 p^{6} T^{18} + 236553 p^{7} T^{19} + 6357 p^{8} T^{20} - 2792 p^{9} T^{21} - 123 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24}
71 (1+246T2+864T3+27087T4+163296T5+2111380T6+163296pT7+27087p2T8+864p3T9+246p4T10+p6T12)2 ( 1 + 246 T^{2} + 864 T^{3} + 27087 T^{4} + 163296 T^{5} + 2111380 T^{6} + 163296 p T^{7} + 27087 p^{2} T^{8} + 864 p^{3} T^{9} + 246 p^{4} T^{10} + p^{6} T^{12} )^{2}
73 (1−12T+282T2−2326T3+34542T4−226674T5+2873211T6−226674pT7+34542p2T8−2326p3T9+282p4T10−12p5T11+p6T12)2 ( 1 - 12 T + 282 T^{2} - 2326 T^{3} + 34542 T^{4} - 226674 T^{5} + 2873211 T^{6} - 226674 p T^{7} + 34542 p^{2} T^{8} - 2326 p^{3} T^{9} + 282 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2}
79 1+24T+75T2−1586T3+9156T4+425598T5+2207908T6−21534534T7−220820679T8+1920297382T9+32423493453T10−57642789732T11−2974107069509T12−57642789732pT13+32423493453p2T14+1920297382p3T15−220820679p4T16−21534534p5T17+2207908p6T18+425598p7T19+9156p8T20−1586p9T21+75p10T22+24p11T23+p12T24 1 + 24 T + 75 T^{2} - 1586 T^{3} + 9156 T^{4} + 425598 T^{5} + 2207908 T^{6} - 21534534 T^{7} - 220820679 T^{8} + 1920297382 T^{9} + 32423493453 T^{10} - 57642789732 T^{11} - 2974107069509 T^{12} - 57642789732 p T^{13} + 32423493453 p^{2} T^{14} + 1920297382 p^{3} T^{15} - 220820679 p^{4} T^{16} - 21534534 p^{5} T^{17} + 2207908 p^{6} T^{18} + 425598 p^{7} T^{19} + 9156 p^{8} T^{20} - 1586 p^{9} T^{21} + 75 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24}
83 1+6T−375T2−1512T3+81582T4+207186T5−12797956T6−19690128T7+1584961659T8+1269102870T9−164608491789T10−478472652pT11+14685582800691T12−478472652p2T13−164608491789p2T14+1269102870p3T15+1584961659p4T16−19690128p5T17−12797956p6T18+207186p7T19+81582p8T20−1512p9T21−375p10T22+6p11T23+p12T24 1 + 6 T - 375 T^{2} - 1512 T^{3} + 81582 T^{4} + 207186 T^{5} - 12797956 T^{6} - 19690128 T^{7} + 1584961659 T^{8} + 1269102870 T^{9} - 164608491789 T^{10} - 478472652 p T^{11} + 14685582800691 T^{12} - 478472652 p^{2} T^{13} - 164608491789 p^{2} T^{14} + 1269102870 p^{3} T^{15} + 1584961659 p^{4} T^{16} - 19690128 p^{5} T^{17} - 12797956 p^{6} T^{18} + 207186 p^{7} T^{19} + 81582 p^{8} T^{20} - 1512 p^{9} T^{21} - 375 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
89 (1−9T+309T2−3006T3+49110T4−463734T5+5132383T6−463734pT7+49110p2T8−3006p3T9+309p4T10−9p5T11+p6T12)2 ( 1 - 9 T + 309 T^{2} - 3006 T^{3} + 49110 T^{4} - 463734 T^{5} + 5132383 T^{6} - 463734 p T^{7} + 49110 p^{2} T^{8} - 3006 p^{3} T^{9} + 309 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
97 1−21T−213T2+4318T3+79581T4−852789T5−15536735T6+93324870T7+2428778718T8−6590604818T9−309431633322T10+305897810664T11+31210720717540T12+305897810664pT13−309431633322p2T14−6590604818p3T15+2428778718p4T16+93324870p5T17−15536735p6T18−852789p7T19+79581p8T20+4318p9T21−213p10T22−21p11T23+p12T24 1 - 21 T - 213 T^{2} + 4318 T^{3} + 79581 T^{4} - 852789 T^{5} - 15536735 T^{6} + 93324870 T^{7} + 2428778718 T^{8} - 6590604818 T^{9} - 309431633322 T^{10} + 305897810664 T^{11} + 31210720717540 T^{12} + 305897810664 p T^{13} - 309431633322 p^{2} T^{14} - 6590604818 p^{3} T^{15} + 2428778718 p^{4} T^{16} + 93324870 p^{5} T^{17} - 15536735 p^{6} T^{18} - 852789 p^{7} T^{19} + 79581 p^{8} T^{20} + 4318 p^{9} T^{21} - 213 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24}
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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

−3.37238969029074383001250513887, −3.23498525246339654134147386759, −3.23095991775238003884677667742, −3.09648674052432573159586412908, −3.06024297758643636099376008926, −2.93538699010000698528484654027, −2.87362636714463341065238374735, −2.79088006352780943250900595982, −2.77988233808895844721951262178, −2.66641481971031820711569172482, −2.65419853693720718136540110051, −2.31203922840097218731575869544, −2.05655809411473315253779068792, −2.00563436551980738890601090915, −1.88325628242928916777574349095, −1.79351611327971733277131532124, −1.77773585070207320184904259370, −1.52510346913649049204251592041, −1.27366781146112692728748362775, −0.967272969664452184742312483763, −0.904576978184203258710874057489, −0.861854829186206914209836506245, −0.852842285448942225754693816618, −0.55846019360582081506188357467, −0.03729868617224404585428071870, 0.03729868617224404585428071870, 0.55846019360582081506188357467, 0.852842285448942225754693816618, 0.861854829186206914209836506245, 0.904576978184203258710874057489, 0.967272969664452184742312483763, 1.27366781146112692728748362775, 1.52510346913649049204251592041, 1.77773585070207320184904259370, 1.79351611327971733277131532124, 1.88325628242928916777574349095, 2.00563436551980738890601090915, 2.05655809411473315253779068792, 2.31203922840097218731575869544, 2.65419853693720718136540110051, 2.66641481971031820711569172482, 2.77988233808895844721951262178, 2.79088006352780943250900595982, 2.87362636714463341065238374735, 2.93538699010000698528484654027, 3.06024297758643636099376008926, 3.09648674052432573159586412908, 3.23095991775238003884677667742, 3.23498525246339654134147386759, 3.37238969029074383001250513887

Graph of the ZZ-function along the critical line

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