L(s) = 1 | + 8·2-s − 18·3-s + 16·4-s − 108·5-s − 144·6-s − 128·8-s + 81·9-s − 864·10-s − 124·11-s − 288·12-s + 1.44e3·13-s + 1.94e3·15-s − 1.02e3·16-s + 1.26e3·17-s + 648·18-s − 360·19-s − 1.72e3·20-s − 992·22-s − 6.52e3·23-s + 2.30e3·24-s + 6.71e3·25-s + 1.15e4·26-s + 1.45e3·27-s + 1.41e4·29-s + 1.55e4·30-s − 5.90e3·31-s − 2.04e3·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.93·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s − 2.73·10-s − 0.308·11-s − 0.577·12-s + 2.36·13-s + 2.23·15-s − 16-s + 1.05·17-s + 0.471·18-s − 0.228·19-s − 0.965·20-s − 0.436·22-s − 2.57·23-s + 0.816·24-s + 2.14·25-s + 3.34·26-s + 0.384·27-s + 3.13·29-s + 3.15·30-s − 1.10·31-s − 0.353·32-s + ⋯ |
Λ(s)=(=((24⋅34⋅78)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((24⋅34⋅78)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅34⋅78
|
Sign: |
1
|
Analytic conductor: |
4.94346×106 |
Root analytic conductor: |
6.86679 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅34⋅78, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
1.473636926 |
L(21) |
≈ |
1.473636926 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | (1−p2T+p4T2)2 |
| 3 | C2 | (1+p2T+p4T2)2 |
| 7 | | 1 |
good | 5 | D4×C2 | 1+108T+4948T2+50328T3−1110969T4+50328p5T5+4948p10T6+108p15T7+p20T8 |
| 11 | D4×C2 | 1+124T−278818T2−3460592T3+58136364859T4−3460592p5T5−278818p10T6+124p15T7+p20T8 |
| 13 | D4 | (1−720T+673736T2−720p5T3+p10T4)2 |
| 17 | D4×C2 | 1−1260T−1209092T2+54207720T3+3551327269215T4+54207720p5T5−1209092p10T6−1260p15T7+p20T8 |
| 19 | D4×C2 | 1+360T−4647630T2−62988480T3+16369957784699T4−62988480p5T5−4647630p10T6+360p15T7+p20T8 |
| 23 | D4×C2 | 1+6524T+19335014T2+2937183088pT3+423716043763p2T4+2937183088p6T5+19335014p10T6+6524p15T7+p20T8 |
| 29 | D4 | (1−7088T+48216146T2−7088p5T3+p10T4)2 |
| 31 | D4×C2 | 1+5904T−23971190T2+9269894016T3+1643220565216419T4+9269894016p5T5−23971190p10T6+5904p15T7+p20T8 |
| 37 | D4×C2 | 1−6040T−74621402T2+166612868480T3+5005514179302955T4+166612868480p5T5−74621402p10T6−6040p15T7+p20T8 |
| 41 | D4 | (1+17388T+216792980T2+17388p5T3+p10T4)2 |
| 43 | D4 | (1+608T+164053110T2+608p5T3+p10T4)2 |
| 47 | D4×C2 | 1−648pT+298147738T2−110633159232pT3+130837578639633603T4−110633159232p6T5+298147738p10T6−648p16T7+p20T8 |
| 53 | D4×C2 | 1+3964T−578718526T2−959126126096T3+171890491073351707T4−959126126096p5T5−578718526p10T6+3964p15T7+p20T8 |
| 59 | D4×C2 | 1+40752T−180305678T2+16756512663168T3+1690986043017054027T4+16756512663168p5T5−180305678p10T6+40752p15T7+p20T8 |
| 61 | D4×C2 | 1−1368T−1543594872T2+196617586608T3+1673542562621074391T4+196617586608p5T5−1543594872p10T6−1368p15T7+p20T8 |
| 67 | D4×C2 | 1−16224T−549961574T2+30615831207936T3−1516953966778043781T4+30615831207936p5T5−549961574p10T6−16224p15T7+p20T8 |
| 71 | D4 | (1+3204T−212201462T2+3204p5T3+p10T4)2 |
| 73 | D4×C2 | 1+23976T+506144208T2−97760673100368T3−5484607792703379793T4−97760673100368p5T5+506144208p10T6+23976p15T7+p20T8 |
| 79 | D4×C2 | 1−1040pT−1035403070T2−1696818106880pT3+30377412596963147299T4−1696818106880p6T5−1035403070p10T6−1040p16T7+p20T8 |
| 83 | D4 | (1−173736T+14732805782T2−173736p5T3+p10T4)2 |
| 89 | D4×C2 | 1+200556T+19064326876T2+2003607258829272T3+19⋯35T4+2003607258829272p5T5+19064326876p10T6+200556p15T7+p20T8 |
| 97 | D4 | (1−251928T+32348722272T2−251928p5T3+p10T4)2 |
show more | | |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.63111350276851282839529306859, −7.58033851108019026727045520910, −6.87432692669783178625408153198, −6.85144687383743232668091520067, −6.46747154302176764150305047309, −6.23335823569125001920137255739, −6.02922364482775022172083533449, −5.81885558169677874373478035254, −5.69018738939597738078411138806, −5.05909528642798878902088862602, −4.88470861381797690758866630384, −4.75816470690841193528849070244, −4.50958013114299861744887360528, −4.04853548720884862749081578565, −3.81783796757596073178725100313, −3.48574176551513432328104320130, −3.44172828692050778258738652739, −3.28653292807623554903241064502, −2.65557472819995092016172701871, −2.16989529920840359056549155876, −1.83577275211195568490860382596, −1.09198641351113775959050559666, −0.978045514965047791858217735623, −0.55655967611740390127177039663, −0.19472807488905400749879755380,
0.19472807488905400749879755380, 0.55655967611740390127177039663, 0.978045514965047791858217735623, 1.09198641351113775959050559666, 1.83577275211195568490860382596, 2.16989529920840359056549155876, 2.65557472819995092016172701871, 3.28653292807623554903241064502, 3.44172828692050778258738652739, 3.48574176551513432328104320130, 3.81783796757596073178725100313, 4.04853548720884862749081578565, 4.50958013114299861744887360528, 4.75816470690841193528849070244, 4.88470861381797690758866630384, 5.05909528642798878902088862602, 5.69018738939597738078411138806, 5.81885558169677874373478035254, 6.02922364482775022172083533449, 6.23335823569125001920137255739, 6.46747154302176764150305047309, 6.85144687383743232668091520067, 6.87432692669783178625408153198, 7.58033851108019026727045520910, 7.63111350276851282839529306859