L(s) = 1 | − 8·2-s − 18·3-s + 16·4-s − 53·5-s + 144·6-s + 128·8-s + 81·9-s + 424·10-s − 191·11-s − 288·12-s + 758·13-s + 954·15-s − 1.02e3·16-s − 340·17-s − 648·18-s − 1.76e3·19-s − 848·20-s + 1.52e3·22-s − 3.23e3·23-s − 2.30e3·24-s + 4.55e3·25-s − 6.06e3·26-s + 1.45e3·27-s + 8.91e3·29-s − 7.63e3·30-s + 1.99e3·31-s + 2.04e3·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.948·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s + 1.34·10-s − 0.475·11-s − 0.577·12-s + 1.24·13-s + 1.09·15-s − 16-s − 0.285·17-s − 0.471·18-s − 1.12·19-s − 0.474·20-s + 0.673·22-s − 1.27·23-s − 0.816·24-s + 1.45·25-s − 1.75·26-s + 0.384·27-s + 1.96·29-s − 1.54·30-s + 0.372·31-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) |
≈ |
0.08090134332 |
L(21) |
≈ |
0.08090134332 |
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+53T−1743T2−89994T3+2176954T4−89994p5T5−1743p10T6+53p15T7+p20T8 |
| 11 | D4×C2 | 1+191T−234735T2−883566pT3+345003244p2T4−883566p6T5−234735p10T6+191p15T7+p20T8 |
| 13 | D4 | (1−379T+488066T2−379p5T3+p10T4)2 |
| 17 | D4×C2 | 1+20pT−1792914T2−18624000pT3+1462296318547T4−18624000p6T5−1792914p10T6+20p16T7+p20T8 |
| 19 | D4×C2 | 1+1769T+1045603T2−5074270360T3−9537626497976T4−5074270360p5T5+1045603p10T6+1769p15T7+p20T8 |
| 23 | D4×C2 | 1+3236T−4980510T2+8347326720T3+129944731805059T4+8347326720p5T5−4980510p10T6+3236p15T7+p20T8 |
| 29 | D4 | (1−4459T+37061638T2−4459p5T3+p10T4)2 |
| 31 | D4×C2 | 1−1994T+10280849T2+126744851310T3−893708155041268T4+126744851310p5T5+10280849p10T6−1994p15T7+p20T8 |
| 37 | D4×C2 | 1+20587T+185423563T2+2052793425004T3+22636782601114654T4+2052793425004p5T5+185423563p10T6+20587p15T7+p20T8 |
| 41 | D4 | (1+8814T+240678562T2+8814p5T3+p10T4)2 |
| 43 | D4 | (1−15853T+338678796T2−15853p5T3+p10T4)2 |
| 47 | D4×C2 | 1−33912T+462240278T2−7769017144224T3+156694760249296899T4−7769017144224p5T5+462240278p10T6−33912p15T7+p20T8 |
| 53 | D4×C2 | 1+49239T+1103481011T2+23861570178636T3+556242294983257398T4+23861570178636p5T5+1103481011p10T6+49239p15T7+p20T8 |
| 59 | D4×C2 | 1+56735T+988331391T2+45426593189460T3+2162900736067650880T4+45426593189460p5T5+988331391p10T6+56735p15T7+p20T8 |
| 61 | D4×C2 | 1−67508T+1731262802T2−76748134547280T3+3424209143194800779T4−76748134547280p5T5+1731262802p10T6−67508p15T7+p20T8 |
| 67 | D4×C2 | 1+75723T+1872823325T2+87906769364370T3+5344056371181586764T4+87906769364370p5T5+1872823325p10T6+75723p15T7+p20T8 |
| 71 | D4 | (1+8992T−681216182T2+8992p5T3+p10T4)2 |
| 73 | D4×C2 | 1+3201T−2300766979T2−5874250509006T3+1021915489991879958T4−5874250509006p5T5−2300766979p10T6+3201p15T7+p20T8 |
| 79 | D4×C2 | 1+26612T−3289450441T2−57387814991556T3+4333754463066999968T4−57387814991556p5T5−3289450441p10T6+26612p15T7+p20T8 |
| 83 | D4 | (1−949T+367057696T2−949p5T3+p10T4)2 |
| 89 | D4×C2 | 1−176562T+12318337010T2−1357352851108032T3+15⋯99T4−1357352851108032p5T5+12318337010p10T6−176562p15T7+p20T8 |
| 97 | D4 | (1−129423T+11942811256T2−129423p5T3+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
−7.60265475787005885523531859905, −7.50662608048168609205794364016, −7.31594869983354040218129889810, −6.96270304934182491780245541264, −6.75756850157364672983408263036, −6.16221680042365728170294524695, −6.11807911051165345421185958271, −6.02060327468357356532514031778, −5.91126277404953145765458783994, −5.04821341870029481448958778208, −4.89996542908997233300932179086, −4.80177503316309585991159397724, −4.56845285496617647628659917294, −4.17146902968841297812886795462, −3.73973190031628130907894059642, −3.42034549365550281877376625769, −3.38478337584057942038021712558, −2.57326640511419660239006376063, −2.49557265211285296202468567236, −1.88723688085889009487778301271, −1.60485374640931317454032050579, −1.08414002896829867392297372945, −0.823118429253004335813725830263, −0.50703506543731530270446824941, −0.092742387068855774462298142356,
0.092742387068855774462298142356, 0.50703506543731530270446824941, 0.823118429253004335813725830263, 1.08414002896829867392297372945, 1.60485374640931317454032050579, 1.88723688085889009487778301271, 2.49557265211285296202468567236, 2.57326640511419660239006376063, 3.38478337584057942038021712558, 3.42034549365550281877376625769, 3.73973190031628130907894059642, 4.17146902968841297812886795462, 4.56845285496617647628659917294, 4.80177503316309585991159397724, 4.89996542908997233300932179086, 5.04821341870029481448958778208, 5.91126277404953145765458783994, 6.02060327468357356532514031778, 6.11807911051165345421185958271, 6.16221680042365728170294524695, 6.75756850157364672983408263036, 6.96270304934182491780245541264, 7.31594869983354040218129889810, 7.50662608048168609205794364016, 7.60265475787005885523531859905