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) |
≈ |
12.87804478 |
L(21) |
≈ |
12.87804478 |
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 |
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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
−7.76098142906066108877135266947, −7.29433711899035757315862484277, −7.08979245573586115360182728680, −6.80950400419099369812627654849, −6.61545259779112006506340454457, −6.13166889015983905626727900124, −5.94286353236680425702889475061, −5.86227944470570397242242344409, −5.74366767397527999003523320786, −5.06355014619340073476629186913, −4.84891800199306761638731825166, −4.72409667515195608625694652029, −4.42972253595802301193796804279, −4.38179243061646611440402596918, −3.68956975617950421620194180398, −3.61278462043272959497986302497, −2.81591379628559282954867351180, −2.79476834306928668934256448948, −2.58413574235771668159232023638, −2.34006760858001028025383182342, −2.15926633740515517148271558999, −1.67471991784309015324619986436, −1.16551238210768898715563918903, −0.64467589223355718018740164248, −0.29408986605787766245766384009,
0.29408986605787766245766384009, 0.64467589223355718018740164248, 1.16551238210768898715563918903, 1.67471991784309015324619986436, 2.15926633740515517148271558999, 2.34006760858001028025383182342, 2.58413574235771668159232023638, 2.79476834306928668934256448948, 2.81591379628559282954867351180, 3.61278462043272959497986302497, 3.68956975617950421620194180398, 4.38179243061646611440402596918, 4.42972253595802301193796804279, 4.72409667515195608625694652029, 4.84891800199306761638731825166, 5.06355014619340073476629186913, 5.74366767397527999003523320786, 5.86227944470570397242242344409, 5.94286353236680425702889475061, 6.13166889015983905626727900124, 6.61545259779112006506340454457, 6.80950400419099369812627654849, 7.08979245573586115360182728680, 7.29433711899035757315862484277, 7.76098142906066108877135266947