L(s) = 1 | + 5-s + 2·7-s + 6·13-s + 14·17-s + 14·19-s − 7·23-s − 6·29-s − 3·31-s + 2·35-s − 12·37-s − 4·41-s − 8·43-s + 4·47-s + 7·49-s − 10·53-s − 6·59-s + 3·61-s + 6·65-s + 10·67-s + 24·71-s + 32·73-s − 79-s − 9·83-s + 14·85-s − 8·89-s + 12·91-s + 14·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.66·13-s + 3.39·17-s + 3.21·19-s − 1.45·23-s − 1.11·29-s − 0.538·31-s + 0.338·35-s − 1.97·37-s − 0.624·41-s − 1.21·43-s + 0.583·47-s + 49-s − 1.37·53-s − 0.781·59-s + 0.384·61-s + 0.744·65-s + 1.22·67-s + 2.84·71-s + 3.74·73-s − 0.112·79-s − 0.987·83-s + 1.51·85-s − 0.847·89-s + 1.25·91-s + 1.43·95-s + ⋯ |
Λ(s)=(=(10497600s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(10497600s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
10497600
= 26⋅38⋅52
|
Sign: |
1
|
Analytic conductor: |
669.336 |
Root analytic conductor: |
5.08640 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 10497600, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
5.331738795 |
L(21) |
≈ |
5.331738795 |
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 | C2 | 1−T+T2 |
good | 7 | C22 | 1−2T−3T2−2pT3+p2T4 |
| 11 | C22 | 1−pT2+p2T4 |
| 13 | C22 | 1−6T+23T2−6pT3+p2T4 |
| 17 | C2 | (1−7T+pT2)2 |
| 19 | C2 | (1−7T+pT2)2 |
| 23 | C22 | 1+7T+26T2+7pT3+p2T4 |
| 29 | C22 | 1+6T+7T2+6pT3+p2T4 |
| 31 | C22 | 1+3T−22T2+3pT3+p2T4 |
| 37 | C2 | (1+6T+pT2)2 |
| 41 | C22 | 1+4T−25T2+4pT3+p2T4 |
| 43 | C2 | (1−5T+pT2)(1+13T+pT2) |
| 47 | C22 | 1−4T−31T2−4pT3+p2T4 |
| 53 | C2 | (1+5T+pT2)2 |
| 59 | C22 | 1+6T−23T2+6pT3+p2T4 |
| 61 | C22 | 1−3T−52T2−3pT3+p2T4 |
| 67 | C22 | 1−10T+33T2−10pT3+p2T4 |
| 71 | C2 | (1−12T+pT2)2 |
| 73 | C2 | (1−16T+pT2)2 |
| 79 | C22 | 1+T−78T2+pT3+p2T4 |
| 83 | C22 | 1+9T−2T2+9pT3+p2T4 |
| 89 | C2 | (1+4T+pT2)2 |
| 97 | C22 | 1−16T+159T2−16pT3+p2T4 |
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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
−8.645056597312782358062323748869, −8.451659412786848327973492643791, −7.938982820141644406833992380808, −7.889871179447226501993421457074, −7.37468764244695724052393142993, −7.18217105161627753759527882209, −6.55422510419272755035961748222, −6.06027471198198289740930481213, −5.65771761174981089982074949307, −5.40800091009319753371250765394, −5.22574979274172579226159946548, −4.90041531420908580183324751915, −3.84390215677984779011606653403, −3.62419632611300659058531839053, −3.30909039291830617980713266152, −3.21314298034999413343180849806, −1.95005600693802729060898423601, −1.83557380284143849680891655418, −1.06672505350067614836791233750, −0.903346765991348911470250951074,
0.903346765991348911470250951074, 1.06672505350067614836791233750, 1.83557380284143849680891655418, 1.95005600693802729060898423601, 3.21314298034999413343180849806, 3.30909039291830617980713266152, 3.62419632611300659058531839053, 3.84390215677984779011606653403, 4.90041531420908580183324751915, 5.22574979274172579226159946548, 5.40800091009319753371250765394, 5.65771761174981089982074949307, 6.06027471198198289740930481213, 6.55422510419272755035961748222, 7.18217105161627753759527882209, 7.37468764244695724052393142993, 7.889871179447226501993421457074, 7.938982820141644406833992380808, 8.451659412786848327973492643791, 8.645056597312782358062323748869