L(s) = 1 | + 8·7-s − 10·9-s + 132·17-s − 264·23-s − 25·25-s + 304·31-s + 876·41-s − 408·47-s − 638·49-s − 80·63-s − 864·71-s − 724·73-s − 320·79-s − 629·81-s − 1.62e3·89-s + 2.21e3·97-s − 2.07e3·113-s + 1.05e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.32e3·153-s + 157-s + ⋯ |
L(s) = 1 | + 0.431·7-s − 0.370·9-s + 1.88·17-s − 2.39·23-s − 1/5·25-s + 1.76·31-s + 3.33·41-s − 1.26·47-s − 1.86·49-s − 0.159·63-s − 1.44·71-s − 1.16·73-s − 0.455·79-s − 0.862·81-s − 1.92·89-s + 2.31·97-s − 1.72·113-s + 0.813·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.697·153-s + 0.000508·157-s + ⋯ |
Λ(s)=(=(1638400s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(1638400s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1638400
= 216⋅52
|
Sign: |
1
|
Analytic conductor: |
5703.63 |
Root analytic conductor: |
8.69036 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1638400, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
2.433466473 |
L(21) |
≈ |
2.433466473 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C2 | 1+p2T2 |
good | 3 | C22 | 1+10T2+p6T4 |
| 7 | C2 | (1−4T+p3T2)2 |
| 11 | C22 | 1−2518T2+p6T4 |
| 13 | C22 | 1−1030T2+p6T4 |
| 17 | C2 | (1−66T+p3T2)2 |
| 19 | C22 | 1−3718T2+p6T4 |
| 23 | C2 | (1+132T+p3T2)2 |
| 29 | C22 | 1−40678T2+p6T4 |
| 31 | C2 | (1−152T+p3T2)2 |
| 37 | C22 | 1−100150T2+p6T4 |
| 41 | C2 | (1−438T+p3T2)2 |
| 43 | C22 | 1−157990T2+p6T4 |
| 47 | C2 | (1+204T+p3T2)2 |
| 53 | C22 | 1−248470T2+p6T4 |
| 59 | C22 | 1−234358T2+p6T4 |
| 61 | C22 | 1+359642T2+p6T4 |
| 67 | C22 | 1+447050T2+p6T4 |
| 71 | C2 | (1+432T+p3T2)2 |
| 73 | C2 | (1+362T+p3T2)2 |
| 79 | C2 | (1+160T+p3T2)2 |
| 83 | C22 | 1−1138390T2+p6T4 |
| 89 | C2 | (1+810T+p3T2)2 |
| 97 | C2 | (1−1106T+p3T2)2 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.922209730572301062251873999987, −9.258253447272109001920048394046, −8.390003232217146344175375369858, −8.310445717514221102856939796935, −7.964986002609789878754630899586, −7.46739943291890699445640228728, −7.32260100981730608015028871905, −6.30180583583198474534969304251, −6.25028994028918074745659101015, −5.70296392808869601463585874376, −5.53260172383680491013297982608, −4.63203746638670448118391121580, −4.46099514712251998840508123575, −3.96525022484581501747524334019, −3.20304566533653100763421543424, −2.98284293069613874450983070645, −2.25801337125959235671303844648, −1.65595392011314236820540630010, −1.09443746676811413137401110765, −0.39504804746985530018064498754,
0.39504804746985530018064498754, 1.09443746676811413137401110765, 1.65595392011314236820540630010, 2.25801337125959235671303844648, 2.98284293069613874450983070645, 3.20304566533653100763421543424, 3.96525022484581501747524334019, 4.46099514712251998840508123575, 4.63203746638670448118391121580, 5.53260172383680491013297982608, 5.70296392808869601463585874376, 6.25028994028918074745659101015, 6.30180583583198474534969304251, 7.32260100981730608015028871905, 7.46739943291890699445640228728, 7.964986002609789878754630899586, 8.310445717514221102856939796935, 8.390003232217146344175375369858, 9.258253447272109001920048394046, 9.922209730572301062251873999987