L(s) = 1 | + 3·2-s − 2·3-s + 4·4-s + 4·5-s − 6·6-s + 11·7-s + 4·8-s − 4·9-s + 12·10-s − 8·12-s + 7·13-s + 33·14-s − 8·15-s + 2·16-s + 3·17-s − 12·18-s + 12·19-s + 16·20-s − 22·21-s − 9·23-s − 8·24-s + 10·25-s + 21·26-s + 11·27-s + 44·28-s − 8·29-s − 24·30-s + ⋯ |
L(s) = 1 | + 2.12·2-s − 1.15·3-s + 2·4-s + 1.78·5-s − 2.44·6-s + 4.15·7-s + 1.41·8-s − 4/3·9-s + 3.79·10-s − 2.30·12-s + 1.94·13-s + 8.81·14-s − 2.06·15-s + 1/2·16-s + 0.727·17-s − 2.82·18-s + 2.75·19-s + 3.57·20-s − 4.80·21-s − 1.87·23-s − 1.63·24-s + 2·25-s + 4.11·26-s + 2.11·27-s + 8.31·28-s − 1.48·29-s − 4.38·30-s + ⋯ |
Λ(s)=(=((54⋅118)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((54⋅118)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
54⋅118
|
Sign: |
1
|
Analytic conductor: |
544.665 |
Root analytic conductor: |
2.19794 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 54⋅118, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
16.30570261 |
L(21) |
≈ |
16.30570261 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 5 | C1 | (1−T)4 |
| 11 | | 1 |
good | 2 | (((C4×C2):C2):C2):C2 | 1−3T+5T2−7T3+11T4−7pT5+5p2T6−3p3T7+p4T8 |
| 3 | C2≀C2≀C2 | 1+2T+8T2+13T3+35T4+13pT5+8p2T6+2p3T7+p4T8 |
| 7 | C2≀C2≀C2 | 1−11T+67T2−276T3+845T4−276pT5+67p2T6−11p3T7+p4T8 |
| 13 | C2≀C2≀C2 | 1−7T+59T2−264T3+1185T4−264pT5+59p2T6−7p3T7+p4T8 |
| 17 | C2≀C2≀C2 | 1−3T+60T2−127T3+1451T4−127pT5+60p2T6−3p3T7+p4T8 |
| 19 | C2≀C2≀C2 | 1−12T+122T2−749T3+3939T4−749pT5+122p2T6−12p3T7+p4T8 |
| 23 | C2≀C2≀C2 | 1+9T+38T2−85T3−979T4−85pT5+38p2T6+9p3T7+p4T8 |
| 29 | C2≀C2≀C2 | 1+8T+112T2+601T3+4759T4+601pT5+112p2T6+8p3T7+p4T8 |
| 31 | C2≀C2≀C2 | 1−3T+3pT2−276T3+3945T4−276pT5+3p3T6−3p3T7+p4T8 |
| 37 | C2≀C2≀C2 | 1+3T+92T2+305T3+4221T4+305pT5+92p2T6+3p3T7+p4T8 |
| 41 | C2≀C2≀C2 | 1+7T+118T2+479T3+5815T4+479pT5+118p2T6+7p3T7+p4T8 |
| 43 | C2≀C2≀C2 | 1−21T+293T2−2900T3+21441T4−2900pT5+293p2T6−21p3T7+p4T8 |
| 47 | C2≀C2≀C2 | 1+3T+137T2+290T3+8531T4+290pT5+137p2T6+3p3T7+p4T8 |
| 53 | C2≀C2≀C2 | 1+11T+245T2+1776T3+20351T4+1776pT5+245p2T6+11p3T7+p4T8 |
| 59 | C2≀C2≀C2 | 1+7T+127T2+44T3+4999T4+44pT5+127p2T6+7p3T7+p4T8 |
| 61 | C2≀C2≀C2 | 1−4T+203T2−642T3+17379T4−642pT5+203p2T6−4p3T7+p4T8 |
| 67 | C2≀C2≀C2 | 1+T+186T2−37T3+15845T4−37pT5+186p2T6+p3T7+p4T8 |
| 71 | C2≀C2≀C2 | 1+15T+186T2+925T3+8531T4+925pT5+186p2T6+15p3T7+p4T8 |
| 73 | C2≀C2≀C2 | 1+9T+218T2+1375T3+20781T4+1375pT5+218p2T6+9p3T7+p4T8 |
| 79 | C2≀C2≀C2 | 1−6T+220T2−487T3+20123T4−487pT5+220p2T6−6p3T7+p4T8 |
| 83 | C2≀C2≀C2 | 1−15T+375T2−3710T3+48443T4−3710pT5+375p2T6−15p3T7+p4T8 |
| 89 | C2≀C2≀C2 | 1+206T2−400T3+21551T4−400pT5+206p2T6+p4T8 |
| 97 | C2≀C2≀C2 | 1−6T+332T2−1836T3+45565T4−1836pT5+332p2T6−6p3T7+p4T8 |
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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
−7.60935323589162439937982439355, −7.44807360774094599999828725945, −7.37948948187619310636626378580, −6.59981833869442945804463241546, −6.58529880887731459114525705933, −6.10107983838590632936608903311, −5.91438795048445607648537405342, −5.82211887530809714242873864901, −5.75307391154236322311450590435, −5.33322061232929930404373761811, −5.17594273225296416952173819251, −5.08869954135693275926803764032, −5.05390119079528486129013001736, −4.56782242226678921392832011731, −4.30527704672202159909961329509, −4.07944597503032509139738913172, −3.86494629608583165998401050357, −3.19930343251278315123318221028, −3.13269866437963588658453419520, −2.82302347031775843151032153474, −2.08303021296616042965844033710, −1.95688892379934227145845556046, −1.58891951458805395883110651479, −1.41060291378352900714376948884, −0.948431305586601174813472817241,
0.948431305586601174813472817241, 1.41060291378352900714376948884, 1.58891951458805395883110651479, 1.95688892379934227145845556046, 2.08303021296616042965844033710, 2.82302347031775843151032153474, 3.13269866437963588658453419520, 3.19930343251278315123318221028, 3.86494629608583165998401050357, 4.07944597503032509139738913172, 4.30527704672202159909961329509, 4.56782242226678921392832011731, 5.05390119079528486129013001736, 5.08869954135693275926803764032, 5.17594273225296416952173819251, 5.33322061232929930404373761811, 5.75307391154236322311450590435, 5.82211887530809714242873864901, 5.91438795048445607648537405342, 6.10107983838590632936608903311, 6.58529880887731459114525705933, 6.59981833869442945804463241546, 7.37948948187619310636626378580, 7.44807360774094599999828725945, 7.60935323589162439937982439355