L(s) = 1 | − 2·3-s + 2·9-s + 12·11-s − 8·23-s + 8·25-s − 6·27-s − 24·33-s − 16·37-s − 32·47-s + 16·49-s + 4·59-s + 16·61-s + 16·69-s − 8·71-s − 8·73-s − 16·75-s + 11·81-s + 20·83-s − 16·97-s + 24·99-s − 12·107-s − 16·109-s + 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 2/3·9-s + 3.61·11-s − 1.66·23-s + 8/5·25-s − 1.15·27-s − 4.17·33-s − 2.63·37-s − 4.66·47-s + 16/7·49-s + 0.520·59-s + 2.04·61-s + 1.92·69-s − 0.949·71-s − 0.936·73-s − 1.84·75-s + 11/9·81-s + 2.19·83-s − 1.62·97-s + 2.41·99-s − 1.16·107-s − 1.53·109-s + 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Λ(s)=(=((228⋅34)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((228⋅34)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
228⋅34
|
Sign: |
1
|
Analytic conductor: |
88.3961 |
Root analytic conductor: |
1.75107 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 228⋅34, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.368055916 |
L(21) |
≈ |
1.368055916 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C22 | 1+2T+2T2+2pT3+p2T4 |
good | 5 | D4×C2 | 1−8T2+46T4−8p2T6+p4T8 |
| 7 | D4×C2 | 1−16T2+142T4−16p2T6+p4T8 |
| 11 | C4 | (1−6T+26T2−6pT3+p2T4)2 |
| 13 | C22 | (1+6T2+p2T4)2 |
| 17 | D4×C2 | 1−20T2+358T4−20p2T6+p4T8 |
| 19 | D4×C2 | 1−48T2+1118T4−48p2T6+p4T8 |
| 23 | D4 | (1+4T+30T2+4pT3+p2T4)2 |
| 29 | D4×C2 | 1−8T2+718T4−8p2T6+p4T8 |
| 31 | D4×C2 | 1−96T2+4046T4−96p2T6+p4T8 |
| 37 | D4 | (1+8T+70T2+8pT3+p2T4)2 |
| 41 | D4×C2 | 1−116T2+6406T4−116p2T6+p4T8 |
| 43 | D4×C2 | 1−112T2+6334T4−112p2T6+p4T8 |
| 47 | C2 | (1+8T+pT2)4 |
| 53 | D4×C2 | 1−200T2+15598T4−200p2T6+p4T8 |
| 59 | D4 | (1−2T+114T2−2pT3+p2T4)2 |
| 61 | D4 | (1−8T+118T2−8pT3+p2T4)2 |
| 67 | D4×C2 | 1−160T2+13758T4−160p2T6+p4T8 |
| 71 | D4 | (1+4T−34T2+4pT3+p2T4)2 |
| 73 | C2 | (1+2T+pT2)4 |
| 79 | D4×C2 | 1−128T2+7758T4−128p2T6+p4T8 |
| 83 | D4 | (1−10T+186T2−10pT3+p2T4)2 |
| 89 | C22 | (1−162T2+p2T4)2 |
| 97 | D4 | (1+8T+190T2+8pT3+p2T4)2 |
show more | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.160166407138318214300651917701, −8.036084077565844897833599344961, −7.85984494523370398543395636615, −7.18525770703991326931854786251, −7.11598315438957270778129682342, −6.79698648521432253349227155231, −6.61433965144405487198754339378, −6.55095056031657267483702077158, −6.40602062376300254341970292162, −5.93826331989077339506237106707, −5.70087326337340059556178010635, −5.38207263894208148420383547763, −5.11666821562140884642663997519, −4.98790155860994296026882195668, −4.38230861728745604736417430792, −4.26104731231808745258099638198, −3.86457554916008562357222693846, −3.74678272242361289903181249756, −3.52229369124652651480767571762, −3.03819750642008649545686305237, −2.52866925710650984195584796654, −1.83548621488853150549222289154, −1.51012220631758179412520442508, −1.45472451603873987349616811025, −0.52833814300405234554427011740,
0.52833814300405234554427011740, 1.45472451603873987349616811025, 1.51012220631758179412520442508, 1.83548621488853150549222289154, 2.52866925710650984195584796654, 3.03819750642008649545686305237, 3.52229369124652651480767571762, 3.74678272242361289903181249756, 3.86457554916008562357222693846, 4.26104731231808745258099638198, 4.38230861728745604736417430792, 4.98790155860994296026882195668, 5.11666821562140884642663997519, 5.38207263894208148420383547763, 5.70087326337340059556178010635, 5.93826331989077339506237106707, 6.40602062376300254341970292162, 6.55095056031657267483702077158, 6.61433965144405487198754339378, 6.79698648521432253349227155231, 7.11598315438957270778129682342, 7.18525770703991326931854786251, 7.85984494523370398543395636615, 8.036084077565844897833599344961, 8.160166407138318214300651917701