L(s) = 1 | − 5-s + 6·9-s + 25-s − 4·29-s + 12·41-s − 6·45-s − 2·49-s + 4·61-s + 27·81-s − 12·89-s + 12·101-s − 28·109-s − 6·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 2·9-s + 1/5·25-s − 0.742·29-s + 1.87·41-s − 0.894·45-s − 2/7·49-s + 0.512·61-s + 3·81-s − 1.27·89-s + 1.19·101-s − 2.68·109-s − 0.545·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
Λ(s)=(=(32000s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(32000s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
32000
= 28⋅53
|
Sign: |
1
|
Analytic conductor: |
2.04034 |
Root analytic conductor: |
1.19515 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 32000, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.340249496 |
L(21) |
≈ |
1.340249496 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C1 | 1+T |
good | 3 | C2 | (1−pT2)2 |
| 7 | C22 | 1+2T2+p2T4 |
| 11 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 13 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 17 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 19 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 23 | C22 | 1−30T2+p2T4 |
| 29 | C2 | (1+2T+pT2)2 |
| 31 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 37 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 41 | C2 | (1−6T+pT2)2 |
| 43 | C22 | 1−22T2+p2T4 |
| 47 | C22 | 1−78T2+p2T4 |
| 53 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 59 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 61 | C2 | (1−2T+pT2)2 |
| 67 | C22 | 1−70T2+p2T4 |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 79 | C2 | (1+pT2)2 |
| 83 | C22 | 1+90T2+p2T4 |
| 89 | C2 | (1+6T+pT2)2 |
| 97 | C2 | (1−14T+pT2)(1+14T+pT2) |
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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
−10.45635555895123398508616992264, −9.932807738951225402896384329207, −9.440574375670347063720129211679, −9.067838966976968249797003880438, −8.176970332316499283761017873268, −7.76702746734900361151457450826, −7.20137597437398225607662382241, −6.85444202581638370159310745997, −6.12074514811380938963071814274, −5.36548914818550921723419960819, −4.57309836509969959885811026572, −4.14489874143322088321845955968, −3.51023978075085753955621242562, −2.34824788381240391505005522266, −1.26603240808509178112653247284,
1.26603240808509178112653247284, 2.34824788381240391505005522266, 3.51023978075085753955621242562, 4.14489874143322088321845955968, 4.57309836509969959885811026572, 5.36548914818550921723419960819, 6.12074514811380938963071814274, 6.85444202581638370159310745997, 7.20137597437398225607662382241, 7.76702746734900361151457450826, 8.176970332316499283761017873268, 9.067838966976968249797003880438, 9.440574375670347063720129211679, 9.932807738951225402896384329207, 10.45635555895123398508616992264