L(s) = 1 | − 3·5-s − 6·17-s − 6·19-s − 30·23-s + 9·25-s − 16·31-s + 48·47-s + 137·49-s + 192·53-s + 38·61-s + 6·79-s − 288·83-s + 18·85-s + 18·95-s + 18·107-s − 226·109-s + 564·113-s + 90·115-s + 129·121-s − 102·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + ⋯ |
L(s) = 1 | − 3/5·5-s − 0.352·17-s − 0.315·19-s − 1.30·23-s + 9/25·25-s − 0.516·31-s + 1.02·47-s + 2.79·49-s + 3.62·53-s + 0.622·61-s + 6/79·79-s − 3.46·83-s + 0.211·85-s + 0.189·95-s + 0.168·107-s − 2.07·109-s + 4.99·113-s + 0.782·115-s + 1.06·121-s − 0.815·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.309·155-s + ⋯ |
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s)4L(s)Λ(3−s)
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s+1)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅312⋅54
|
Sign: |
1
|
Analytic conductor: |
1.19992×107 |
Root analytic conductor: |
7.67174 |
Motivic weight: |
2 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅312⋅54, ( :1,1,1,1), 1)
|
Particular Values
L(23) |
≈ |
2.888538046 |
L(21) |
≈ |
2.888538046 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C22 | 1+3T+3p2T3+p4T4 |
good | 7 | D4×C2 | 1−137T2+9024T4−137p4T6+p8T8 |
| 11 | D4×C2 | 1−129T2+10400T4−129p4T6+p8T8 |
| 13 | D4×C2 | 1−136T2−5970T4−136p4T6+p8T8 |
| 17 | D4 | (1+3T+528T2+3p2T3+p4T4)2 |
| 19 | D4 | (1+3T+254T2+3p2T3+p4T4)2 |
| 23 | D4 | (1+15T+1062T2+15p2T3+p4T4)2 |
| 29 | C22 | (1−98T2+p4T4)2 |
| 31 | D4 | (1+8T+57T2+8p2T3+p4T4)2 |
| 37 | C22 | (1−1522T2+p4T4)2 |
| 41 | C22 | (1−3186T2+p4T4)2 |
| 43 | D4×C2 | 1−776T2+5716590T4−776p4T6+p8T8 |
| 47 | D4 | (1−24T+3726T2−24p2T3+p4T4)2 |
| 53 | D4 | (1−96T+6041T2−96p2T3+p4T4)2 |
| 59 | D4×C2 | 1−7044T2+28934630T4−7044p4T6+p8T8 |
| 61 | D4 | (1−19T+7062T2−19p2T3+p4T4)2 |
| 67 | D4×C2 | 1−6584T2+19350606T4−6584p4T6+p8T8 |
| 71 | D4×C2 | 1−3208T2+50078094T4−3208p4T6+p8T8 |
| 73 | D4×C2 | 1−9521T2+50769360T4−9521p4T6+p8T8 |
| 79 | D4 | (1−3T+8252T2−3p2T3+p4T4)2 |
| 83 | D4 | (1+144T+13737T2+144p2T3+p4T4)2 |
| 89 | D4×C2 | 1−6984T2+4586510T4−6984p4T6+p8T8 |
| 97 | D4×C2 | 1+1879T2+54186000T4+1879p4T6+p8T8 |
show more | | |
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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
−6.45275063207953615304089024260, −5.79164533082923924781390514747, −5.79008241202683818535708973968, −5.74215812804999647912029325241, −5.48222760952389867665591201241, −5.24645140823958225849295368585, −5.11701133734558483652206175100, −4.68241730709267918121942179489, −4.33827998672569627841733873404, −4.33426453395391530260449804978, −4.12483819882266062802836670877, −3.83751839180455988459629681127, −3.77733451259936906185745775943, −3.62249088879339623771724542396, −3.01140912643776800838049716719, −2.92262854989699602723186920788, −2.60865609810213438551997379665, −2.48003195240326275318872101856, −2.11813636678865547521278669505, −1.84655260032679177117511499079, −1.69763018371610120969033403761, −1.07857019234062449529969872411, −0.880902383580385417073733515747, −0.55606554905684952155676777865, −0.25989186106014136577233636213,
0.25989186106014136577233636213, 0.55606554905684952155676777865, 0.880902383580385417073733515747, 1.07857019234062449529969872411, 1.69763018371610120969033403761, 1.84655260032679177117511499079, 2.11813636678865547521278669505, 2.48003195240326275318872101856, 2.60865609810213438551997379665, 2.92262854989699602723186920788, 3.01140912643776800838049716719, 3.62249088879339623771724542396, 3.77733451259936906185745775943, 3.83751839180455988459629681127, 4.12483819882266062802836670877, 4.33426453395391530260449804978, 4.33827998672569627841733873404, 4.68241730709267918121942179489, 5.11701133734558483652206175100, 5.24645140823958225849295368585, 5.48222760952389867665591201241, 5.74215812804999647912029325241, 5.79008241202683818535708973968, 5.79164533082923924781390514747, 6.45275063207953615304089024260