L(s) = 1 | + 3-s + 3·4-s + 3·5-s + 9-s + 9·11-s + 3·12-s − 14·13-s + 3·15-s + 4·16-s − 12·17-s − 9·19-s + 9·20-s + 12·23-s − 3·25-s − 4·27-s + 9·29-s + 30·31-s + 9·33-s + 3·36-s − 14·39-s − 18·41-s − 5·43-s + 27·44-s + 3·45-s + 24·47-s + 4·48-s − 12·51-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 3/2·4-s + 1.34·5-s + 1/3·9-s + 2.71·11-s + 0.866·12-s − 3.88·13-s + 0.774·15-s + 16-s − 2.91·17-s − 2.06·19-s + 2.01·20-s + 2.50·23-s − 3/5·25-s − 0.769·27-s + 1.67·29-s + 5.38·31-s + 1.56·33-s + 1/2·36-s − 2.24·39-s − 2.81·41-s − 0.762·43-s + 4.07·44-s + 0.447·45-s + 3.50·47-s + 0.577·48-s − 1.68·51-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) |
≈ |
6.307219756 |
L(21) |
≈ |
6.307219756 |
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 | C22×C22 | (1−T−T2−pT3+p2T4)(1+T−T2+pT3+p2T4) |
| 3 | D4×C2 | 1−T+5T3−11T4+5pT5−p3T7+p4T8 |
| 5 | D4×C2 | 1−3T+12T2−27T3+71T4−27pT5+12p2T6−3p3T7+p4T8 |
| 11 | D4×C2 | 1−9T+54T2−243T3+905T4−243pT5+54p2T6−9p3T7+p4T8 |
| 17 | C2 | (1+3T+pT2)4 |
| 19 | D4×C2 | 1+9T+56T2+261T3+993T4+261pT5+56p2T6+9p3T7+p4T8 |
| 23 | D4 | (1−6T+34T2−6pT3+p2T4)2 |
| 29 | D4×C2 | 1−9T+8T2−135T3+2139T4−135pT5+8p2T6−9p3T7+p4T8 |
| 31 | C2 | (1−11T+pT2)2(1−4T+pT2)2 |
| 37 | C2 | (1−10T+pT2)2(1+10T+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 | 1−6T2+5627T4−6p2T6+p4T8 |
| 61 | D4×C2 | 1+2T+70T2−376T3+391T4−376pT5+70p2T6+2p3T7+p4T8 |
| 67 | C2×C22 | (1+4T+pT2)2(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 | 1−87T2+1853T4−87p2T6+p4T8 |
| 97 | D4×C2 | 1−39T+812T2−11895T3+132795T4−11895pT5+812p2T6−39p3T7+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
−7.41797706601794476924588567162, −7.37334985932552469135494613465, −7.07829446431214611821892572457, −6.73026388398757019681449655512, −6.56139384277718308642843366832, −6.45740792489184069708560120282, −6.42883799761839721886692724428, −6.39050970461606434915508643514, −5.91958148762236628856279437785, −5.28170885232712290420499610630, −5.16368294417713953081998970209, −4.94130153665576688602015884051, −4.56559258033460906841430322472, −4.32948954678276299851584239643, −4.22720016079006004795848593384, −4.18422274463419537881382882272, −3.29720734956952129651356291656, −3.10690378896087616522524560119, −2.65797190259664534910305746339, −2.43295887203806432238362271193, −2.26865212919591692733733551697, −2.20527056451832008666077908539, −1.78838542180004913324572604439, −1.24420679570262469178396602579, −0.63068482637995740321499257054,
0.63068482637995740321499257054, 1.24420679570262469178396602579, 1.78838542180004913324572604439, 2.20527056451832008666077908539, 2.26865212919591692733733551697, 2.43295887203806432238362271193, 2.65797190259664534910305746339, 3.10690378896087616522524560119, 3.29720734956952129651356291656, 4.18422274463419537881382882272, 4.22720016079006004795848593384, 4.32948954678276299851584239643, 4.56559258033460906841430322472, 4.94130153665576688602015884051, 5.16368294417713953081998970209, 5.28170885232712290420499610630, 5.91958148762236628856279437785, 6.39050970461606434915508643514, 6.42883799761839721886692724428, 6.45740792489184069708560120282, 6.56139384277718308642843366832, 6.73026388398757019681449655512, 7.07829446431214611821892572457, 7.37334985932552469135494613465, 7.41797706601794476924588567162