L(s) = 1 | + 18·3-s + 36·5-s + 120·7-s + 243·9-s + 200·11-s + 284·13-s + 648·15-s + 2.67e3·17-s − 72·19-s + 2.16e3·21-s − 3.84e3·23-s + 2.65e3·25-s + 2.91e3·27-s + 1.02e4·29-s − 1.04e4·31-s + 3.60e3·33-s + 4.32e3·35-s + 1.31e4·37-s + 5.11e3·39-s + 4.16e3·41-s + 5.83e3·43-s + 8.74e3·45-s − 1.52e3·47-s − 1.48e4·49-s + 4.81e4·51-s + 9.01e3·53-s + 7.20e3·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.643·5-s + 0.925·7-s + 9-s + 0.498·11-s + 0.466·13-s + 0.743·15-s + 2.24·17-s − 0.0457·19-s + 1.06·21-s − 1.51·23-s + 0.850·25-s + 0.769·27-s + 2.25·29-s − 1.96·31-s + 0.575·33-s + 0.596·35-s + 1.57·37-s + 0.538·39-s + 0.386·41-s + 0.481·43-s + 0.643·45-s − 0.100·47-s − 0.885·49-s + 2.59·51-s + 0.440·53-s + 0.320·55-s + ⋯ |
Λ(s)=(=(9216s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(9216s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
9216
= 210⋅32
|
Sign: |
1
|
Analytic conductor: |
237.062 |
Root analytic conductor: |
3.92388 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 9216, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
6.411855281 |
L(21) |
≈ |
6.411855281 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | (1−p2T)2 |
good | 5 | D4 | 1−36T−1362T2−36p5T3+p10T4 |
| 7 | D4 | 1−120T+29278T2−120p5T3+p10T4 |
| 11 | D4 | 1−200T+46406T2−200p5T3+p10T4 |
| 13 | D4 | 1−284T+477054T2−284p5T3+p10T4 |
| 17 | D4 | 1−2676T+4344262T2−2676p5T3+p10T4 |
| 19 | D4 | 1+72T+4159894T2+72p5T3+p10T4 |
| 23 | D4 | 1+3840T+9416686T2+3840p5T3+p10T4 |
| 29 | D4 | 1−10212T+64228638T2−10212p5T3+p10T4 |
| 31 | D4 | 1+10488T+84559438T2+10488p5T3+p10T4 |
| 37 | D4 | 1−13148T+125908974T2−13148p5T3+p10T4 |
| 41 | D4 | 1−4164T+216207126T2−4164p5T3+p10T4 |
| 43 | D4 | 1−5832T+168401542T2−5832p5T3+p10T4 |
| 47 | D4 | 1+1520T+148144670T2+1520p5T3+p10T4 |
| 53 | D4 | 1−9012T+816689646T2−9012p5T3+p10T4 |
| 59 | D4 | 1−55096T+1818478886T2−55096p5T3+p10T4 |
| 61 | D4 | 1+63444T+2677193342T2+63444p5T3+p10T4 |
| 67 | D4 | 1+36792T+1148752246T2+36792p5T3+p10T4 |
| 71 | D4 | 1+37664T+2440628942T2+37664p5T3+p10T4 |
| 73 | D4 | 1+37836T+2085902966T2+37836p5T3+p10T4 |
| 79 | D4 | 1+144888T+10711615534T2+144888p5T3+p10T4 |
| 83 | D4 | 1−109272T+10472055958T2−109272p5T3+p10T4 |
| 89 | D4 | 1+32556T+8836559958T2+32556p5T3+p10T4 |
| 97 | D4 | 1−69092T+18339537030T2−69092p5T3+p10T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.31474362258981328664402008445, −12.91680540892338424814573793409, −12.10912481332933460893494155649, −11.94009229481645761521227229090, −11.04682137455119076810370795351, −10.42098276578784588577857983225, −9.913199136562547562677378010591, −9.546549543467672285070756495215, −8.626802183863672773537982939927, −8.505875212030996092364174976014, −7.54457031083119041491321289681, −7.50694669914831601736673824687, −6.24329802029815766213454453775, −5.84140299252974182624626090693, −4.90464331602273466484279221205, −4.17550519339418360688206196839, −3.37453894067809688386766629978, −2.61657655144675772129530448179, −1.60951289621691539682008742832, −1.09472059715368022048047907437,
1.09472059715368022048047907437, 1.60951289621691539682008742832, 2.61657655144675772129530448179, 3.37453894067809688386766629978, 4.17550519339418360688206196839, 4.90464331602273466484279221205, 5.84140299252974182624626090693, 6.24329802029815766213454453775, 7.50694669914831601736673824687, 7.54457031083119041491321289681, 8.505875212030996092364174976014, 8.626802183863672773537982939927, 9.546549543467672285070756495215, 9.913199136562547562677378010591, 10.42098276578784588577857983225, 11.04682137455119076810370795351, 11.94009229481645761521227229090, 12.10912481332933460893494155649, 12.91680540892338424814573793409, 13.31474362258981328664402008445