L(s) = 1 | + 2·2-s + 4-s + 2·5-s + 3·7-s − 2·8-s + 4·10-s − 5·11-s + 6·14-s − 4·16-s − 3·17-s − 8·19-s + 2·20-s − 10·22-s − 4·23-s − 11·25-s + 3·28-s + 8·29-s − 2·32-s − 6·34-s + 6·35-s − 11·37-s − 16·38-s − 4·40-s + 6·43-s − 5·44-s − 8·46-s + 4·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s − 0.707·8-s + 1.26·10-s − 1.50·11-s + 1.60·14-s − 16-s − 0.727·17-s − 1.83·19-s + 0.447·20-s − 2.13·22-s − 0.834·23-s − 2.19·25-s + 0.566·28-s + 1.48·29-s − 0.353·32-s − 1.02·34-s + 1.01·35-s − 1.80·37-s − 2.59·38-s − 0.632·40-s + 0.914·43-s − 0.753·44-s − 1.17·46-s + 0.583·47-s + ⋯ |
Λ(s)=(=((24⋅312⋅134)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((24⋅312⋅134)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅312⋅134
|
Sign: |
1
|
Analytic conductor: |
987.317 |
Root analytic conductor: |
2.36759 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅312⋅134, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.125159550 |
L(21) |
≈ |
1.125159550 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | (1−T+T2)2 |
| 3 | | 1 |
| 13 | C22 | 1+pT2+p2T4 |
good | 5 | D4 | (1−T+7T2−pT3+p2T4)2 |
| 7 | D4×C2 | 1−3T−4T2+3T3+57T4+3pT5−4p2T6−3p3T7+p4T8 |
| 11 | D4×C2 | 1+5T+15T3+229T4+15pT5+5p3T7+p4T8 |
| 17 | D4×C2 | 1+3T+2T2−81T3−393T4−81pT5+2p2T6+3p3T7+p4T8 |
| 19 | D4×C2 | 1+8T+23T2+24T3+104T4+24pT5+23p2T6+8p3T7+p4T8 |
| 23 | D4×C2 | 1+4T−21T2−36T3+472T4−36pT5−21p2T6+4p3T7+p4T8 |
| 29 | D4×C2 | 1−8T+3T2−24T3+1024T4−24pT5+3p2T6−8p3T7+p4T8 |
| 31 | C22 | (1+10T2+p2T4)2 |
| 37 | D4×C2 | 1+11T+20T2+297T3+4355T4+297pT5+20p2T6+11p3T7+p4T8 |
| 41 | C23 | 1+35T2−456T4+35p2T6+p4T8 |
| 43 | D4×C2 | 1−6T−46T2+24T3+3327T4+24pT5−46p2T6−6p3T7+p4T8 |
| 47 | D4 | (1−2T+82T2−2pT3+p2T4)2 |
| 53 | C2 | (1−3T+pT2)4 |
| 59 | D4×C2 | 1+25T+6pT2+3825T3+33085T4+3825pT5+6p3T6+25p3T7+p4T8 |
| 61 | D4×C2 | 1+15T+50T2+795T3+13179T4+795pT5+50p2T6+15p3T7+p4T8 |
| 67 | D4×C2 | 1−9T−70T2−153T3+12885T4−153pT5−70p2T6−9p3T7+p4T8 |
| 71 | D4×C2 | 1+17T+78T2+1173T3+18961T4+1173pT5+78p2T6+17p3T7+p4T8 |
| 73 | D4 | (1−13T+159T2−13pT3+p2T4)2 |
| 79 | D4 | (1−7T+141T2−7pT3+p2T4)2 |
| 83 | C2 | (1+pT2)4 |
| 89 | D4×C2 | 1+12T+47T2−972T3−11328T4−972pT5+47p2T6+12p3T7+p4T8 |
| 97 | D4×C2 | 1−2T−74T2+232T3−3713T4+232pT5−74p2T6−2p3T7+p4T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.44931710155077606701573428231, −7.39336877213513045991341390559, −7.04978983255978034289662667015, −6.53038239759625173192531350412, −6.45083983970373677243994309033, −6.32283785767617856387873362745, −5.83683665687215791952549977896, −5.78118850307618146031729945867, −5.77762306615379169465255777428, −5.27786471653958429536602100601, −5.02963971157197880670249197722, −4.91945304089873147997206047390, −4.63187556954007289097846632475, −4.35951409709449011408945432981, −4.13405019361891599841001095803, −3.83411503509831759783413542754, −3.78131676387272725834947081726, −3.15902558402276383427883431045, −2.94295667457664057063841697905, −2.36715786900079416652273695234, −2.25012810494293363524167302470, −2.23222473214889527123274306491, −1.69428987950209548240702949921, −1.24149652574381936140579056937, −0.20582211314878541047355333241,
0.20582211314878541047355333241, 1.24149652574381936140579056937, 1.69428987950209548240702949921, 2.23222473214889527123274306491, 2.25012810494293363524167302470, 2.36715786900079416652273695234, 2.94295667457664057063841697905, 3.15902558402276383427883431045, 3.78131676387272725834947081726, 3.83411503509831759783413542754, 4.13405019361891599841001095803, 4.35951409709449011408945432981, 4.63187556954007289097846632475, 4.91945304089873147997206047390, 5.02963971157197880670249197722, 5.27786471653958429536602100601, 5.77762306615379169465255777428, 5.78118850307618146031729945867, 5.83683665687215791952549977896, 6.32283785767617856387873362745, 6.45083983970373677243994309033, 6.53038239759625173192531350412, 7.04978983255978034289662667015, 7.39336877213513045991341390559, 7.44931710155077606701573428231