L(s) = 1 | + 6·2-s + 21·4-s − 5-s + 2·7-s + 56·8-s − 6·10-s + 11-s + 8·13-s + 12·14-s + 126·16-s + 4·17-s − 3·19-s − 21·20-s + 6·22-s + 7·23-s + 9·25-s + 48·26-s + 42·28-s + 5·29-s − 40·31-s + 252·32-s + 24·34-s − 2·35-s + 3·37-s − 18·38-s − 56·40-s − 6·43-s + ⋯ |
L(s) = 1 | + 4.24·2-s + 21/2·4-s − 0.447·5-s + 0.755·7-s + 19.7·8-s − 1.89·10-s + 0.301·11-s + 2.21·13-s + 3.20·14-s + 63/2·16-s + 0.970·17-s − 0.688·19-s − 4.69·20-s + 1.27·22-s + 1.45·23-s + 9/5·25-s + 9.41·26-s + 7.93·28-s + 0.928·29-s − 7.18·31-s + 44.5·32-s + 4.11·34-s − 0.338·35-s + 0.493·37-s − 2.91·38-s − 8.85·40-s − 0.914·43-s + ⋯ |
Λ(s)=(=((26⋅318⋅76)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((26⋅318⋅76)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
66.07648851 |
L(21) |
≈ |
66.07648851 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | (1−T)6 |
| 3 | 1 |
| 7 | 1−2T−4T2+31T3−4pT4−2p2T5+p3T6 |
good | 5 | 1+T−8T2−17T3+23T4+52T5−11T6+52pT7+23p2T8−17p3T9−8p4T10+p5T11+p6T12 |
| 11 | 1−T−26T2+23T3+37pT4−202T5−4853T6−202pT7+37p3T8+23p3T9−26p4T10−p5T11+p6T12 |
| 13 | 1−8T+24T2−42T3−32T4+1408T5−7901T6+1408pT7−32p2T8−42p3T9+24p4T10−8p5T11+p6T12 |
| 17 | 1−4T−23T2+4pT3+410T4−220T5−8111T6−220pT7+410p2T8+4p4T9−23p4T10−4p5T11+p6T12 |
| 19 | 1+3T−12T2−67T3−153T4+54T5+6315T6+54pT7−153p2T8−67p3T9−12p4T10+3p5T11+p6T12 |
| 23 | 1−7T−32T2+83T3+2423T4−3946T5−46865T6−3946pT7+2423p2T8+83p3T9−32p4T10−7p5T11+p6T12 |
| 29 | 1−5T+4T2−251T3+197T4+3418T5+20293T6+3418pT7+197p2T8−251p3T9+4p4T10−5p5T11+p6T12 |
| 31 | (1+20T+214T2+1441T3+214pT4+20p2T5+p3T6)2 |
| 37 | (1−11T+pT2)3(1+10T+pT2)3 |
| 41 | 1−90T2−18T3+4410T4+810T5−194177T6+810pT7+4410p2T8−18p3T9−90p4T10+p6T12 |
| 43 | (1−12T−6T2+547T3−6pT4−12p2T5+p3T6)(1+18T+198T2+1519T3+198pT4+18p2T5+p3T6) |
| 47 | (1+9T+87T2+657T3+87pT4+9p2T5+p3T6)2 |
| 53 | 1+15T+33T3+13635T4+60360T5−225155T6+60360pT7+13635p2T8+33p3T9+15p5T11+p6T12 |
| 59 | (1+14T+216T2+1589T3+216pT4+14p2T5+p3T6)2 |
| 61 | (1+8T+178T2+883T3+178pT4+8p2T5+p3T6)2 |
| 67 | (1+T+89T2−77T3+89pT4+p2T5+p3T6)2 |
| 71 | (1+7T+15T2−599T3+15pT4+7p2T5+p3T6)2 |
| 73 | 1−19T+134T2−27T3−5759T4+41986T5−314903T6+41986pT7−5759p2T8−27p3T9+134p4T10−19p5T11+p6T12 |
| 79 | (1+5T+163T2+469T3+163pT4+5p2T5+p3T6)2 |
| 83 | 1+2T−182T2+2T3+18788T4−13564T5−1721225T6−13564pT7+18788p2T8+2p3T9−182p4T10+2p5T11+p6T12 |
| 89 | 1−9T−144T2+1197T3+16101T4−73314T5−1141967T6−73314pT7+16101p2T8+1197p3T9−144p4T10−9p5T11+p6T12 |
| 97 | 1−28T+281T2−2724T3+45178T4−388196T5+2169217T6−388196pT7+45178p2T8−2724p3T9+281p4T10−28p5T11+p6T12 |
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L(s)=p∏ j=1∏12(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.07878265916212801829735997984, −6.02501738194667865983584268696, −5.66501125091915023492950419213, −5.32562995097282969507034760154, −5.32152919242022192799525238896, −5.30677659557406930733845350369, −5.27025091890013245020229281948, −4.73029169813143067734238008664, −4.66724574913227276869726483909, −4.47046179352367386985837698871, −4.33193110203303504909978094090, −4.26923249519894237669821071818, −3.94746548860829451773895114774, −3.55586632460255318295741375386, −3.34584628962931518742568875030, −3.29951961768226093161864040708, −3.20316648994883505590536208853, −3.16534886381249362879113209807, −3.10888205063618535068314362421, −2.36429474107570020497919387375, −1.92114892921075885986819548267, −1.75608942409201619663298673545, −1.73428687845461665640461730786, −1.57367847758576328833735746604, −0.933281394689002347831166355662,
0.933281394689002347831166355662, 1.57367847758576328833735746604, 1.73428687845461665640461730786, 1.75608942409201619663298673545, 1.92114892921075885986819548267, 2.36429474107570020497919387375, 3.10888205063618535068314362421, 3.16534886381249362879113209807, 3.20316648994883505590536208853, 3.29951961768226093161864040708, 3.34584628962931518742568875030, 3.55586632460255318295741375386, 3.94746548860829451773895114774, 4.26923249519894237669821071818, 4.33193110203303504909978094090, 4.47046179352367386985837698871, 4.66724574913227276869726483909, 4.73029169813143067734238008664, 5.27025091890013245020229281948, 5.30677659557406930733845350369, 5.32152919242022192799525238896, 5.32562995097282969507034760154, 5.66501125091915023492950419213, 6.02501738194667865983584268696, 6.07878265916212801829735997984
Plot not available for L-functions of degree greater than 10.