L(s) = 1 | + 2·2-s + 3-s − 2·4-s − 5-s + 2·6-s − 7·8-s − 10·9-s − 2·10-s + 4·11-s − 2·12-s − 15-s − 4·16-s + 5·17-s − 20·18-s + 19-s + 2·20-s + 8·22-s + 23-s − 7·24-s − 18·25-s − 10·27-s − 3·29-s − 2·30-s − 16·31-s + 4·32-s + 4·33-s + 10·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s − 4-s − 0.447·5-s + 0.816·6-s − 2.47·8-s − 3.33·9-s − 0.632·10-s + 1.20·11-s − 0.577·12-s − 0.258·15-s − 16-s + 1.21·17-s − 4.71·18-s + 0.229·19-s + 0.447·20-s + 1.70·22-s + 0.208·23-s − 1.42·24-s − 3.59·25-s − 1.92·27-s − 0.557·29-s − 0.365·30-s − 2.87·31-s + 0.707·32-s + 0.696·33-s + 1.71·34-s + ⋯ |
Λ(s)=(=((712⋅1312)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((712⋅1312)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.090704165 |
L(21) |
≈ |
1.090704165 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1 |
good | 2 | 1−pT+3pT2−9T3+5p2T4−7p2T5+51T6−7p3T7+5p4T8−9p3T9+3p5T10−p6T11+p6T12 |
| 3 | 1−T+11T2−11T3+19pT4−55T5+197T6−55pT7+19p3T8−11p3T9+11p4T10−p5T11+p6T12 |
| 5 | 1+T+19T2+7T3+161T4−3T5+913T6−3pT7+161p2T8+7p3T9+19p4T10+p5T11+p6T12 |
| 11 | 1−4T+45T2−144T3+972T4−2539T5+13237T6−2539pT7+972p2T8−144p3T9+45p4T10−4p5T11+p6T12 |
| 17 | 1−5T+90T2−411T3+3539T4−13744T5+78123T6−13744pT7+3539p2T8−411p3T9+90p4T10−5p5T11+p6T12 |
| 19 | 1−T+50T2+16T3+1180T4+1175T5+23331T6+1175pT7+1180p2T8+16p3T9+50p4T10−p5T11+p6T12 |
| 23 | 1−T+32T2−52T3+1214T4−1383T5+21935T6−1383pT7+1214p2T8−52p3T9+32p4T10−p5T11+p6T12 |
| 29 | 1+3T+96T2+191T3+4061T4+5126T5+122643T6+5126pT7+4061p2T8+191p3T9+96p4T10+3p5T11+p6T12 |
| 31 | 1+16T+236T2+2185T3+18573T4+122325T5+755039T6+122325pT7+18573p2T8+2185p3T9+236p4T10+16p5T11+p6T12 |
| 37 | 1+13T+184T2+1054T3+7158T4+10573T5+113729T6+10573pT7+7158p2T8+1054p3T9+184p4T10+13p5T11+p6T12 |
| 41 | 1−8T+225T2−1362T3+21488T4−101725T5+1145451T6−101725pT7+21488p2T8−1362p3T9+225p4T10−8p5T11+p6T12 |
| 43 | 1−11T+259T2−2099T3+27622T4−170696T5+1576761T6−170696pT7+27622p2T8−2099p3T9+259p4T10−11p5T11+p6T12 |
| 47 | 1−T+105T2−147T3+5543T4−3359T5+246951T6−3359pT7+5543p2T8−147p3T9+105p4T10−p5T11+p6T12 |
| 53 | 1−2T+218T2−344T3+22040T4−26940T5+1409201T6−26940pT7+22040p2T8−344p3T9+218p4T10−2p5T11+p6T12 |
| 59 | 1+13T+5pT2+2839T3+38957T4+294699T5+2963017T6+294699pT7+38957p2T8+2839p3T9+5p5T10+13p5T11+p6T12 |
| 61 | 1+5T+165T2+599T3+15743T4+53393T5+1179159T6+53393pT7+15743p2T8+599p3T9+165p4T10+5p5T11+p6T12 |
| 67 | 1+11T+296T2+2796T3+41197T4+326168T5+3447813T6+326168pT7+41197p2T8+2796p3T9+296p4T10+11p5T11+p6T12 |
| 71 | 1−6T+285T2−14pT3+35468T4−74185T5+2901951T6−74185pT7+35468p2T8−14p4T9+285p4T10−6p5T11+p6T12 |
| 73 | 1−30T+676T2−10699T3+141027T4−1519265T5+14149139T6−1519265pT7+141027p2T8−10699p3T9+676p4T10−30p5T11+p6T12 |
| 79 | 1+7T+326T2+2455T3+54096T4+344443T5+5474643T6+344443pT7+54096p2T8+2455p3T9+326p4T10+7p5T11+p6T12 |
| 83 | 1+27T+656T2+10802T3+153994T4+1760871T5+17670883T6+1760871pT7+153994p2T8+10802p3T9+656p4T10+27p5T11+p6T12 |
| 89 | 1+4T+167T2+1648T3+21035T4+202110T5+2204075T6+202110pT7+21035p2T8+1648p3T9+167p4T10+4p5T11+p6T12 |
| 97 | 1−35T+947T2−18161T3+281670T4−3629766T5+38644781T6−3629766pT7+281670p2T8−18161p3T9+947p4T10−35p5T11+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
−4.02301834815495183224958416038, −3.68241364426194685139385720080, −3.64491283110275752117144736944, −3.61304943038765930289590630884, −3.53835350614457093369441444045, −3.46767223481295891361957026408, −3.29145931385397189884827768778, −3.08016723100250928398057604476, −2.89511292394892601136779838335, −2.83903014321783801968820330978, −2.76329155787451344306404330881, −2.65294708378062721644923515938, −2.23851562029172358502785151207, −2.14240052526736777498970592628, −2.03422250890345436126006244357, −1.96261037083161603607083263733, −1.86246597012376464590779029227, −1.49050829122657866389018035259, −1.42483110475111036611536594729, −1.36699431739397690870270681519, −0.68355653587700443792888520791, −0.65785245355063660305618623073, −0.48406870277325265509681665042, −0.42658562472398187631631534260, −0.10710580178647573481117557401,
0.10710580178647573481117557401, 0.42658562472398187631631534260, 0.48406870277325265509681665042, 0.65785245355063660305618623073, 0.68355653587700443792888520791, 1.36699431739397690870270681519, 1.42483110475111036611536594729, 1.49050829122657866389018035259, 1.86246597012376464590779029227, 1.96261037083161603607083263733, 2.03422250890345436126006244357, 2.14240052526736777498970592628, 2.23851562029172358502785151207, 2.65294708378062721644923515938, 2.76329155787451344306404330881, 2.83903014321783801968820330978, 2.89511292394892601136779838335, 3.08016723100250928398057604476, 3.29145931385397189884827768778, 3.46767223481295891361957026408, 3.53835350614457093369441444045, 3.61304943038765930289590630884, 3.64491283110275752117144736944, 3.68241364426194685139385720080, 4.02301834815495183224958416038
Plot not available for L-functions of degree greater than 10.