L(s) = 1 | + 2·5-s − 4·7-s − 2·11-s + 8·19-s + 8·23-s + 3·25-s − 4·29-s + 4·31-s − 8·35-s − 20·37-s − 4·41-s + 12·43-s + 8·47-s + 4·49-s + 4·53-s − 4·55-s + 4·59-s − 4·61-s + 8·67-s + 12·71-s − 16·73-s + 8·77-s + 16·79-s + 20·83-s + 12·89-s + 16·95-s + 4·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.603·11-s + 1.83·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s + 0.718·31-s − 1.35·35-s − 3.28·37-s − 0.624·41-s + 1.82·43-s + 1.16·47-s + 4/7·49-s + 0.549·53-s − 0.539·55-s + 0.520·59-s − 0.512·61-s + 0.977·67-s + 1.42·71-s − 1.87·73-s + 0.911·77-s + 1.80·79-s + 2.19·83-s + 1.27·89-s + 1.64·95-s + 0.406·97-s + ⋯ |
Λ(s)=(=(15681600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(15681600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
15681600
= 26⋅34⋅52⋅112
|
Sign: |
1
|
Analytic conductor: |
999.872 |
Root analytic conductor: |
5.62323 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 15681600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.858976627 |
L(21) |
≈ |
2.858976627 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C1 | (1−T)2 |
| 11 | C1 | (1+T)2 |
good | 7 | D4 | 1+4T+12T2+4pT3+p2T4 |
| 13 | C22 | 1+20T2+p2T4 |
| 17 | C22 | 1−20T2+p2T4 |
| 19 | C2 | (1−4T+pT2)2 |
| 23 | D4 | 1−8T+38T2−8pT3+p2T4 |
| 29 | D4 | 1+4T+38T2+4pT3+p2T4 |
| 31 | D4 | 1−4T+42T2−4pT3+p2T4 |
| 37 | C2 | (1+10T+pT2)2 |
| 41 | D4 | 1+4T+62T2+4pT3+p2T4 |
| 43 | D4 | 1−12T+116T2−12pT3+p2T4 |
| 47 | D4 | 1−8T+86T2−8pT3+p2T4 |
| 53 | D4 | 1−4T+86T2−4pT3+p2T4 |
| 59 | D4 | 1−4T+98T2−4pT3+p2T4 |
| 61 | D4 | 1+4T+102T2+4pT3+p2T4 |
| 67 | D4 | 1−8T+126T2−8pT3+p2T4 |
| 71 | D4 | 1−12T+154T2−12pT3+p2T4 |
| 73 | D4 | 1+16T+204T2+16pT3+p2T4 |
| 79 | C2 | (1−8T+pT2)2 |
| 83 | D4 | 1−20T+260T2−20pT3+p2T4 |
| 89 | D4 | 1−12T+118T2−12pT3+p2T4 |
| 97 | D4 | 1−4T−18T2−4pT3+p2T4 |
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
−8.688753792895882553474355636060, −8.528633342332084164473615578680, −7.58450179255986974046041419750, −7.57984812212914586957889886155, −7.18001135667186017149315877847, −6.79868852388129303641608044586, −6.42608260458580955634514258103, −6.13329509021090842591414572928, −5.55468616977031845343059360524, −5.37036531021693198767950003250, −5.00324700012710475441794134109, −4.72103364291698952059877680068, −3.77159105777784424880462864900, −3.55887413332452762988172788514, −3.17822917291874172861441739335, −2.84032870176974380774474966247, −2.24106950928213074185701560452, −1.84488793571576616093021777299, −0.953949788459649734628350040479, −0.59123861440622527349243450724,
0.59123861440622527349243450724, 0.953949788459649734628350040479, 1.84488793571576616093021777299, 2.24106950928213074185701560452, 2.84032870176974380774474966247, 3.17822917291874172861441739335, 3.55887413332452762988172788514, 3.77159105777784424880462864900, 4.72103364291698952059877680068, 5.00324700012710475441794134109, 5.37036531021693198767950003250, 5.55468616977031845343059360524, 6.13329509021090842591414572928, 6.42608260458580955634514258103, 6.79868852388129303641608044586, 7.18001135667186017149315877847, 7.57984812212914586957889886155, 7.58450179255986974046041419750, 8.528633342332084164473615578680, 8.688753792895882553474355636060