L(s) = 1 | − 3·2-s + 2·3-s + 3·4-s + 3·5-s − 6·6-s + 9-s − 9·10-s + 6·12-s + 14·13-s + 6·15-s − 2·16-s − 6·17-s − 3·18-s + 9·20-s − 6·23-s − 3·25-s − 42·26-s + 4·27-s + 9·29-s − 18·30-s + 30·31-s − 6·32-s + 18·34-s + 3·36-s − 24·37-s + 28·39-s + 18·41-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 1.15·3-s + 3/2·4-s + 1.34·5-s − 2.44·6-s + 1/3·9-s − 2.84·10-s + 1.73·12-s + 3.88·13-s + 1.54·15-s − 1/2·16-s − 1.45·17-s − 0.707·18-s + 2.01·20-s − 1.25·23-s − 3/5·25-s − 8.23·26-s + 0.769·27-s + 1.67·29-s − 3.28·30-s + 5.38·31-s − 1.06·32-s + 3.08·34-s + 1/2·36-s − 3.94·37-s + 4.48·39-s + 2.81·41-s + ⋯ |
Λ(s)=(=((78⋅134)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((78⋅134)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
78⋅134
|
Sign: |
1
|
Analytic conductor: |
669.369 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 78⋅134, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.064708735 |
L(21) |
≈ |
2.064708735 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 7 | | 1 |
| 13 | C2 | (1−7T+pT2)2 |
good | 2 | C2×C22 | (1+T+pT2)2(1+T−T2+pT3+p2T4) |
| 3 | D4 | (1−T+T2−pT3+p2T4)2 |
| 5 | D4×C2 | 1−3T+12T2−27T3+71T4−27pT5+12p2T6−3p3T7+p4T8 |
| 11 | D4×C2 | 1−27T2+377T4−27p2T6+p4T8 |
| 17 | C22 | (1+3T−8T2+3pT3+p2T4)2 |
| 19 | D4×C2 | 1−31T2+537T4−31p2T6+p4T8 |
| 23 | D4×C2 | 1+6T+2T2−72T3−201T4−72pT5+2p2T6+6p3T7+p4T8 |
| 29 | D4×C2 | 1−9T+8T2−135T3+2139T4−135pT5+8p2T6−9p3T7+p4T8 |
| 31 | C2 | (1−11T+pT2)2(1−4T+pT2)2 |
| 37 | C2 | (1+T+pT2)2(1+11T+pT2)2 |
| 41 | D4×C2 | 1−18T+210T2−1836T3+13151T4−1836pT5+210p2T6−18p3T7+p4T8 |
| 43 | D4×C2 | 1+5T−20T2−205T3−899T4−205pT5−20p2T6+5p3T7+p4T8 |
| 47 | D4×C2 | 1−24T+327T2−3240T3+25040T4−3240pT5+327p2T6−24p3T7+p4T8 |
| 53 | D4×C2 | 1−6T+5T2+450T3−3756T4+450pT5+5p2T6−6p3T7+p4T8 |
| 59 | D4×C2 | 1+6T+21T2+54T3−2692T4+54pT5+21p2T6+6p3T7+p4T8 |
| 61 | D4 | (1+2T−66T2+2pT3+p2T4)2 |
| 67 | C22×C22 | (1−4T−51T2−4pT3+p2T4)(1+4T−51T2+4pT3+p2T4) |
| 71 | D4×C2 | 1−6T+150T2−828T3+14855T4−828pT5+150p2T6−6p3T7+p4T8 |
| 73 | C22 | (1+6T+85T2+6pT3+p2T4)2 |
| 79 | C22 | (1−6T−43T2−6pT3+p2T4)2 |
| 83 | D4×C2 | 1−270T2+31667T4−270p2T6+p4T8 |
| 89 | D4×C2 | 1−33T+588T2−7425T3+75011T4−7425pT5+588p2T6−33p3T7+p4T8 |
| 97 | D4×C2 | 1+39T+812T2+11895T3+132795T4+11895pT5+812p2T6+39p3T7+p4T8 |
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
−8.059837860544257229439088438684, −7.65052726775847318673802253791, −7.01797988008727037178015339679, −6.97267993423042363951242290168, −6.90956719701582355528530522475, −6.31290557489062687614342156901, −6.31027458480504666287269709440, −6.01571060741293661110360917894, −5.84295557175905360746108664247, −5.83910227222597428154607054059, −5.33606203996878601312637995115, −4.77251483431949154101224655195, −4.57006767846061637399974969783, −4.44343398440299428151454173163, −3.99539365037331035321988627286, −3.72216751480771481919384553930, −3.59456460822967411823273910689, −3.01559399350518113244723723505, −2.89304293312412584816532785551, −2.42799224242947469819032395164, −1.99830548317027371002894820121, −1.98924295628334234159921066381, −1.28545071516219709894410587161, −0.941230193954480565229120074302, −0.74887727119996927461514584847,
0.74887727119996927461514584847, 0.941230193954480565229120074302, 1.28545071516219709894410587161, 1.98924295628334234159921066381, 1.99830548317027371002894820121, 2.42799224242947469819032395164, 2.89304293312412584816532785551, 3.01559399350518113244723723505, 3.59456460822967411823273910689, 3.72216751480771481919384553930, 3.99539365037331035321988627286, 4.44343398440299428151454173163, 4.57006767846061637399974969783, 4.77251483431949154101224655195, 5.33606203996878601312637995115, 5.83910227222597428154607054059, 5.84295557175905360746108664247, 6.01571060741293661110360917894, 6.31027458480504666287269709440, 6.31290557489062687614342156901, 6.90956719701582355528530522475, 6.97267993423042363951242290168, 7.01797988008727037178015339679, 7.65052726775847318673802253791, 8.059837860544257229439088438684