L(s) = 1 | − 2·3-s + 6·4-s + 3·5-s + 6·7-s + 9-s + 2·11-s − 12·12-s − 6·13-s − 6·15-s + 12·16-s + 2·17-s − 19-s + 18·20-s − 12·21-s + 3·25-s + 2·27-s + 36·28-s − 7·29-s − 2·31-s − 4·33-s + 18·35-s + 6·36-s + 8·37-s + 12·39-s − 10·41-s − 24·43-s + 12·44-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 3·4-s + 1.34·5-s + 2.26·7-s + 1/3·9-s + 0.603·11-s − 3.46·12-s − 1.66·13-s − 1.54·15-s + 3·16-s + 0.485·17-s − 0.229·19-s + 4.02·20-s − 2.61·21-s + 3/5·25-s + 0.384·27-s + 6.80·28-s − 1.29·29-s − 0.359·31-s − 0.696·33-s + 3.04·35-s + 36-s + 1.31·37-s + 1.92·39-s − 1.56·41-s − 3.65·43-s + 1.80·44-s + ⋯ |
Λ(s)=(=((56⋅76⋅196)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((56⋅76⋅196)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.798874583 |
L(21) |
≈ |
3.798874583 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | (1−T+T2)3 |
| 7 | (1−T)6 |
| 19 | 1+T+20T2+109T3+20pT4+p2T5+p3T6 |
good | 2 | (1−pT2+p2T4)3 |
| 3 | (1+T−2T2−pT3−2pT4+p2T5+p3T6)(1+T+4T2+pT3+4pT4+p2T5+p3T6) |
| 11 | (1−T+12T2−25T3+12pT4−p2T5+p3T6)2 |
| 13 | (1−5T+pT2)3(1+7T+pT2)3 |
| 17 | 1−2T−11T2+154T3−214T4−938T5+13309T6−938pT7−214p2T8+154p3T9−11p4T10−2p5T11+p6T12 |
| 23 | 1−45T2+72T3+990T4−1620T5−20945T6−1620pT7+990p2T8+72p3T9−45p4T10+p6T12 |
| 29 | 1+7T−17T2−104T3+461T4−2231T5−38666T6−2231pT7+461p2T8−104p3T9−17p4T10+7p5T11+p6T12 |
| 31 | (1+T+56T2+133T3+56pT4+p2T5+p3T6)2 |
| 37 | (1−4T+79T2−332T3+79pT4−4p2T5+p3T6)2 |
| 41 | 1+10T−35T2−242T3+4730T4+13570T5−143867T6+13570pT7+4730p2T8−242p3T9−35p4T10+10p5T11+p6T12 |
| 43 | (1−5T+pT2)3(1+13T+pT2)3 |
| 47 | 1−6T−9T2+930T3−3690T4−14874T5+392947T6−14874pT7−3690p2T8+930p3T9−9p4T10−6p5T11+p6T12 |
| 53 | 1−2T−119T2−38T3+8402T4+7666T5−504503T6+7666pT7+8402p2T8−38p3T9−119p4T10−2p5T11+p6T12 |
| 59 | 1+19T+73T2+586T3+19535T4+124027T5+228670T6+124027pT7+19535p2T8+586p3T9+73p4T10+19p5T11+p6T12 |
| 61 | 1−9T−45T2+284T3+1809T4+15525T5−313194T6+15525pT7+1809p2T8+284p3T9−45p4T10−9p5T11+p6T12 |
| 67 | 1+10T+15T2+426T3−590T4−50990T5−286505T6−50990pT7−590p2T8+426p3T9+15p4T10+10p5T11+p6T12 |
| 71 | 1−17T+T2+34T3+20591T4−100385T5−553826T6−100385pT7+20591p2T8+34p3T9+p4T10−17p5T11+p6T12 |
| 73 | 1−4T−127T2+924T3+7018T4−44768T5−240503T6−44768pT7+7018p2T8+924p3T9−127p4T10−4p5T11+p6T12 |
| 79 | 1+7T−159T2−654T3+19447T4+33079T5−1624562T6+33079pT7+19447p2T8−654p3T9−159p4T10+7p5T11+p6T12 |
| 83 | (1+pT2)6 |
| 89 | 1−9T−189T2+756T3+34533T4−68859T5−3182114T6−68859pT7+34533p2T8+756p3T9−189p4T10−9p5T11+p6T12 |
| 97 | 1+8T−67T2−1800T3−6938T4+58504T5+1586329T6+58504pT7−6938p2T8−1800p3T9−67p4T10+8p5T11+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
−5.60297457239646193371860414196, −5.50047444795480605473615821646, −5.12527121961347585086115477722, −5.05564068325907021099691728858, −5.05335202002362261970458141249, −5.04132452468578451201922401357, −4.86865830399438948402640867262, −4.51513710424722095410160159672, −4.26807698214335605336871959369, −4.12867598070293127442270290456, −3.88156633781335282792008990787, −3.70281177899608783973360560580, −3.28831626653203967889496407979, −3.18964867804754293255362322908, −3.01978762615933085619447505044, −2.51900529330654304013893099245, −2.51440157163552321460994915680, −2.47146273661509135958980768859, −2.02994869980025784886523812867, −1.86444768550210045004125809371, −1.83654800738561650800935495607, −1.64414928021479718420897213953, −1.30032700899642871063281104695, −1.12174822512070078898066327636, −0.26882885662505451771965231000,
0.26882885662505451771965231000, 1.12174822512070078898066327636, 1.30032700899642871063281104695, 1.64414928021479718420897213953, 1.83654800738561650800935495607, 1.86444768550210045004125809371, 2.02994869980025784886523812867, 2.47146273661509135958980768859, 2.51440157163552321460994915680, 2.51900529330654304013893099245, 3.01978762615933085619447505044, 3.18964867804754293255362322908, 3.28831626653203967889496407979, 3.70281177899608783973360560580, 3.88156633781335282792008990787, 4.12867598070293127442270290456, 4.26807698214335605336871959369, 4.51513710424722095410160159672, 4.86865830399438948402640867262, 5.04132452468578451201922401357, 5.05335202002362261970458141249, 5.05564068325907021099691728858, 5.12527121961347585086115477722, 5.50047444795480605473615821646, 5.60297457239646193371860414196
Plot not available for L-functions of degree greater than 10.