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 + 6/17·17-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) |
≈ |
2.558189264 |
L(21) |
≈ |
2.558189264 |
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 | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.25463217206279760292280676176, −6.07213391360948891186447392939, −5.92154072264997632557879587901, −5.49380056410662025556531619293, −5.33266981132248607014922667984, −5.28446453240214151900805790271, −5.12956633168430795171826042771, −4.77196822507263347157883499006, −4.55443008358671759128392437993, −4.46434148931892996586207064854, −3.99880228810112708617311903516, −3.90534232403851293415952699847, −3.66835289118221716613950145951, −3.33450483032914835040803969372, −3.09713444608772066687689511379, −3.04831472959587823557731024274, −2.45767643306983300593775718562, −2.37979568795128069548292038516, −2.37766732125980999550645154237, −1.65543679345467919426987487368, −1.48737997847937688239799373727, −1.47463515624593381083772208146, −0.835810586421629522373645965072, −0.68546635599078927958049217665, −0.18879155904879151652914180847,
0.18879155904879151652914180847, 0.68546635599078927958049217665, 0.835810586421629522373645965072, 1.47463515624593381083772208146, 1.48737997847937688239799373727, 1.65543679345467919426987487368, 2.37766732125980999550645154237, 2.37979568795128069548292038516, 2.45767643306983300593775718562, 3.04831472959587823557731024274, 3.09713444608772066687689511379, 3.33450483032914835040803969372, 3.66835289118221716613950145951, 3.90534232403851293415952699847, 3.99880228810112708617311903516, 4.46434148931892996586207064854, 4.55443008358671759128392437993, 4.77196822507263347157883499006, 5.12956633168430795171826042771, 5.28446453240214151900805790271, 5.33266981132248607014922667984, 5.49380056410662025556531619293, 5.92154072264997632557879587901, 6.07213391360948891186447392939, 6.25463217206279760292280676176