L(s) = 1 | − 132·9-s − 1.12e3·11-s − 2.00e3·19-s − 2.68e3·29-s + 4.49e3·31-s + 4.61e4·41-s + 5.65e4·49-s + 1.25e5·59-s + 2.82e4·61-s − 9.44e4·71-s − 1.31e5·79-s − 3.07e4·81-s + 1.10e5·89-s + 1.47e5·99-s + 2.05e5·101-s − 5.57e4·109-s + 1.76e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.73e5·169-s + ⋯ |
L(s) = 1 | − 0.543·9-s − 2.79·11-s − 1.27·19-s − 0.591·29-s + 0.840·31-s + 4.28·41-s + 3.36·49-s + 4.68·59-s + 0.970·61-s − 2.22·71-s − 2.37·79-s − 0.520·81-s + 1.47·89-s + 1.51·99-s + 2.00·101-s − 0.449·109-s + 1.09·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.00·169-s + ⋯ |
Λ(s)=(=((216⋅58)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((216⋅58)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅58
|
Sign: |
1
|
Analytic conductor: |
1.69387×107 |
Root analytic conductor: |
8.00958 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅58, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
0.1485185113 |
L(21) |
≈ |
0.1485185113 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | | 1 |
good | 3 | D4×C2 | 1+44pT2+5350p2T4+44p11T6+p20T8 |
| 7 | D4×C2 | 1−8084pT2+1352943558T4−8084p11T6+p20T8 |
| 11 | D4 | (1+560T+381926T2+560p5T3+p10T4)2 |
| 13 | D4×C2 | 1−373292T2+167403787638T4−373292p10T6+p20T8 |
| 17 | D4×C2 | 1+1613316T2+4602934743878T4+1613316p10T6+p20T8 |
| 19 | D4 | (1+1000T+4533462T2+1000p5T3+p10T4)2 |
| 23 | D4×C2 | 1−7126092T2+48612883261958T4−7126092p10T6+p20T8 |
| 29 | D4 | (1+1340T−8758306T2+1340p5T3+p10T4)2 |
| 31 | D4 | (1−2248T+57017022T2−2248p5T3+p10T4)2 |
| 37 | D4×C2 | 1−211585036T2+19959790722550422T4−211585036p10T6+p20T8 |
| 41 | D4 | (1−23076T+352280470T2−23076p5T3+p10T4)2 |
| 43 | D4×C2 | 1−313073372T2+49182412950657078T4−313073372p10T6+p20T8 |
| 47 | D4×C2 | 1−104863596T2+104529727880710502T4−104863596p10T6+p20T8 |
| 53 | D4×C2 | 1−1357205900T2+805869805574922198T4−1357205900p10T6+p20T8 |
| 59 | D4 | (1−62584T+2277965606T2−62584p5T3+p10T4)2 |
| 61 | D4 | (1−14108T+1110042462T2−14108p5T3+p10T4)2 |
| 67 | D4×C2 | 1−1448611388T2+3060385933795078038T4−1448611388p10T6+p20T8 |
| 71 | D4 | (1+47208T+4011779662T2+47208p5T3+p10T4)2 |
| 73 | D4×C2 | 1−4251788572T2+9098112686809466598T4−4251788572p10T6+p20T8 |
| 79 | D4 | (1+65904T+3994274078T2+65904p5T3+p10T4)2 |
| 83 | D4×C2 | 1−9324010812T2+49682906641812448598T4−9324010812p10T6+p20T8 |
| 89 | D4 | (1−55020T+10818978262T2−55020p5T3+p10T4)2 |
| 97 | D4×C2 | 1−1419021116T2−92174956617487206138T4−1419021116p10T6+p20T8 |
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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
−7.37307672484729644342613798632, −7.19793958871723590049513432859, −6.79641939462733697407627998063, −6.68781920614316502626505503896, −6.21714727104239879680102179070, −5.79422326553415760912951776346, −5.74171955167324646849867698644, −5.63030960014733504128481092149, −5.33147719771114408454969296635, −5.23454829926243579760615188934, −4.50371632290345239577125651660, −4.45389882057029468973322330317, −4.25510654705136452631931707885, −3.97520514206760639738487698235, −3.63285294921867625123270432616, −3.06889897384129707026770485874, −2.85181799238527380361650591663, −2.58056705762740563015287731042, −2.36164130550640247167346062190, −2.17472251969225840768940775764, −1.92542722407541760351906555412, −0.980261759313446434123884768672, −0.858948845768503965434699113946, −0.66491686848561790630722575290, −0.05286702441760970953360181297,
0.05286702441760970953360181297, 0.66491686848561790630722575290, 0.858948845768503965434699113946, 0.980261759313446434123884768672, 1.92542722407541760351906555412, 2.17472251969225840768940775764, 2.36164130550640247167346062190, 2.58056705762740563015287731042, 2.85181799238527380361650591663, 3.06889897384129707026770485874, 3.63285294921867625123270432616, 3.97520514206760639738487698235, 4.25510654705136452631931707885, 4.45389882057029468973322330317, 4.50371632290345239577125651660, 5.23454829926243579760615188934, 5.33147719771114408454969296635, 5.63030960014733504128481092149, 5.74171955167324646849867698644, 5.79422326553415760912951776346, 6.21714727104239879680102179070, 6.68781920614316502626505503896, 6.79641939462733697407627998063, 7.19793958871723590049513432859, 7.37307672484729644342613798632