L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 4·7-s − 2·8-s − 4·10-s − 4·11-s + 2·13-s + 8·14-s − 4·16-s + 12·19-s − 2·20-s − 8·22-s − 2·23-s + 8·25-s + 4·26-s + 4·28-s − 2·29-s + 4·31-s − 2·32-s − 8·35-s − 24·37-s + 24·38-s + 4·40-s − 18·41-s + 4·43-s − 4·44-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 0.707·8-s − 1.26·10-s − 1.20·11-s + 0.554·13-s + 2.13·14-s − 16-s + 2.75·19-s − 0.447·20-s − 1.70·22-s − 0.417·23-s + 8/5·25-s + 0.784·26-s + 0.755·28-s − 0.371·29-s + 0.718·31-s − 0.353·32-s − 1.35·35-s − 3.94·37-s + 3.89·38-s + 0.632·40-s − 2.81·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
Λ(s)=(=((24⋅316⋅134)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((24⋅316⋅134)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅316⋅134
|
Sign: |
1
|
Analytic conductor: |
79972.7 |
Root analytic conductor: |
4.10079 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅316⋅134, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.5260259247 |
L(21) |
≈ |
0.5260259247 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | (1−T+T2)2 |
| 3 | | 1 |
| 13 | C2 | (1−T+T2)2 |
good | 5 | D4×C2 | 1+2T−4T2−4T3+19T4−4pT5−4p2T6+2p3T7+p4T8 |
| 7 | D4×C2 | 1−4T+T2−4T3+64T4−4pT5+p2T6−4p3T7+p4T8 |
| 11 | D4×C2 | 1+4T−7T2+4T3+232T4+4pT5−7p2T6+4p3T7+p4T8 |
| 17 | C22 | (1+22T2+p2T4)2 |
| 19 | D4 | (1−6T+35T2−6pT3+p2T4)2 |
| 23 | D4×C2 | 1+2T−16T2−52T3−221T4−52pT5−16p2T6+2p3T7+p4T8 |
| 29 | D4×C2 | 1+2T−43T2−22T3+1252T4−22pT5−43p2T6+2p3T7+p4T8 |
| 31 | D4×C2 | 1−4T−38T2+32T3+1459T4+32pT5−38p2T6−4p3T7+p4T8 |
| 37 | D4 | (1+12T+98T2+12pT3+p2T4)2 |
| 41 | D4×C2 | 1+18T+4pT2+1404T3+10635T4+1404pT5+4p3T6+18p3T7+p4T8 |
| 43 | D4×C2 | 1−4T−26T2+176T3−773T4+176pT5−26p2T6−4p3T7+p4T8 |
| 47 | D4×C2 | 1+16T+110T2+832T3+7075T4+832pT5+110p2T6+16p3T7+p4T8 |
| 53 | D4 | (1−18T+175T2−18pT3+p2T4)2 |
| 59 | D4×C2 | 1+8T−67T2+104T3+9904T4+104pT5−67p2T6+8p3T7+p4T8 |
| 61 | D4×C2 | 1+8T−47T2−88T3+4696T4−88pT5−47p2T6+8p3T7+p4T8 |
| 67 | D4×C2 | 1+8T+22T2−736T3−7013T4−736pT5+22p2T6+8p3T7+p4T8 |
| 71 | D4 | (1−10T+155T2−10pT3+p2T4)2 |
| 73 | D4 | (1−4T+138T2−4pT3+p2T4)2 |
| 79 | D4×C2 | 1−2T−128T2+52T3+10867T4+52pT5−128p2T6−2p3T7+p4T8 |
| 83 | C23 | 1−163T2+19680T4−163p2T6+p4T8 |
| 89 | D4 | (1−6T+160T2−6pT3+p2T4)2 |
| 97 | D4×C2 | 1+10T−116T2+220T3+27547T4+220pT5−116p2T6+10p3T7+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
−6.65495605911808078002642626910, −6.05505466283440958196458853658, −5.78352058987840913788428577415, −5.70287610151473388827034524293, −5.49504355795238319256167123954, −5.39148317235382564909973808226, −5.04566340429515439781577502537, −4.94221495514448330439039268979, −4.89717722135159697096672964178, −4.75777179193384338369110591872, −4.42104189111532090502491692338, −3.94586966352482256103471520908, −3.81579806839292712127108358467, −3.76438198009230223649727214923, −3.36680999624882043998214294260, −3.27389088621332962054789077218, −3.15778854311640775513978274279, −2.67607569047536963884230594186, −2.50950836345299019447686592868, −2.06104523704225003684701285957, −1.94309141310677753050423118824, −1.39156657826037108155557357547, −1.14509969527452040214592027751, −0.948906439198902314950339057350, −0.093154557555477989788543417803,
0.093154557555477989788543417803, 0.948906439198902314950339057350, 1.14509969527452040214592027751, 1.39156657826037108155557357547, 1.94309141310677753050423118824, 2.06104523704225003684701285957, 2.50950836345299019447686592868, 2.67607569047536963884230594186, 3.15778854311640775513978274279, 3.27389088621332962054789077218, 3.36680999624882043998214294260, 3.76438198009230223649727214923, 3.81579806839292712127108358467, 3.94586966352482256103471520908, 4.42104189111532090502491692338, 4.75777179193384338369110591872, 4.89717722135159697096672964178, 4.94221495514448330439039268979, 5.04566340429515439781577502537, 5.39148317235382564909973808226, 5.49504355795238319256167123954, 5.70287610151473388827034524293, 5.78352058987840913788428577415, 6.05505466283440958196458853658, 6.65495605911808078002642626910