L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 12·7-s + 5·8-s − 2·9-s + 3·11-s + 2·12-s + 6·13-s + 24·14-s + 5·16-s + 7·17-s − 4·18-s + 5·19-s + 12·21-s + 6·22-s + 6·23-s + 5·24-s + 12·26-s + 24·28-s − 15·29-s − 12·31-s − 2·32-s + 3·33-s + 14·34-s − 4·36-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 4.53·7-s + 1.76·8-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 1.66·13-s + 6.41·14-s + 5/4·16-s + 1.69·17-s − 0.942·18-s + 1.14·19-s + 2.61·21-s + 1.27·22-s + 1.25·23-s + 1.02·24-s + 2.35·26-s + 4.53·28-s − 2.78·29-s − 2.15·31-s − 0.353·32-s + 0.522·33-s + 2.40·34-s − 2/3·36-s + ⋯ |
Λ(s)=(=((516)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((516)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
516
|
Sign: |
1
|
Analytic conductor: |
620.338 |
Root analytic conductor: |
2.23397 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 516, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
21.71932009 |
L(21) |
≈ |
21.71932009 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 5 | | 1 |
good | 2 | C22:C4 | 1−pT+pT2−5T3+11T4−5pT5+p3T6−p4T7+p4T8 |
| 3 | C22:C4 | 1−T+pT2−5T3+16T4−5pT5+p3T6−p3T7+p4T8 |
| 7 | C2 | (1−3T+pT2)4 |
| 11 | C4×C2 | 1−3T−2T2+39T3−95T4+39pT5−2p2T6−3p3T7+p4T8 |
| 13 | C22:C4 | 1−6T+23T2−120T3+601T4−120pT5+23p2T6−6p3T7+p4T8 |
| 17 | C22:C4 | 1−7T+2T2+65T3−169T4+65pT5+2p2T6−7p3T7+p4T8 |
| 19 | C22:C4 | 1−5T−9T2+5pT3−184T4+5p2T5−9p2T6−5p3T7+p4T8 |
| 23 | C22:C4 | 1−6T−7T2+30T3+361T4+30pT5−7p2T6−6p3T7+p4T8 |
| 29 | C4×C2 | 1+15T+106T2+675T3+4171T4+675pT5+106p2T6+15p3T7+p4T8 |
| 31 | C22:C4 | 1+12T+63T2+14pT3+105pT4+14p2T5+63p2T6+12p3T7+p4T8 |
| 37 | C22:C4 | 1+18T+107T2+210T3+T4+210pT5+107p2T6+18p3T7+p4T8 |
| 41 | C4×C2 | 1−3T−32T2+219T3+655T4+219pT5−32p2T6−3p3T7+p4T8 |
| 43 | C2 | (1−9T+pT2)4 |
| 47 | C22:C4 | 1+18T+137T2+990T3+7951T4+990pT5+137p2T6+18p3T7+p4T8 |
| 53 | C22:C4 | 1−6T−37T2+360T3+121T4+360pT5−37p2T6−6p3T7+p4T8 |
| 59 | C22:C4 | 1+15T+31T2−15pT3−10424T4−15p2T5+31p2T6+15p3T7+p4T8 |
| 61 | C4×C4 | (1+T−29T2+pT3+p2T4)(1+11T+51T2+11pT3+p2T4) |
| 67 | C22:C4 | 1+18T+77T2−1110T3−17669T4−1110pT5+77p2T6+18p3T7+p4T8 |
| 71 | C4×C2 | 1−3T−62T2+399T3+3205T4+399pT5−62p2T6−3p3T7+p4T8 |
| 73 | C22:C4 | 1+9T−37T2−165T3+4096T4−165pT5−37p2T6+9p3T7+p4T8 |
| 79 | C22:C4 | 1−5T+6T2−715T3+9821T4−715pT5+6p2T6−5p3T7+p4T8 |
| 83 | C22:C4 | 1−6T−7T2−720T3+11221T4−720pT5−7p2T6−6p3T7+p4T8 |
| 89 | C4×C2 | 1+30T+451T2+5400T3+56341T4+5400pT5+451p2T6+30p3T7+p4T8 |
| 97 | C22:C4 | 1−12T−43T2+900T3−1859T4+900pT5−43p2T6−12p3T7+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
−7.64357488615353333304081278006, −7.54874302374735635294936186923, −7.19858437646776198408952664174, −7.17774897889637640638201346750, −6.99743636863288176826705985370, −6.04205954952820237389470179231, −5.95721728587460759671213193035, −5.91446712471854276113862244309, −5.55106444768791377223910327914, −5.54181899346438456754172192226, −4.98054999661851797107399223761, −4.81688616114965219914484008719, −4.74707649054486146018078675739, −4.53966820073703962365294411420, −4.41367792985774835793984553484, −3.66302559260902772792797634801, −3.62986753811717619225181640535, −3.48743510272140338323432857979, −3.34258169756407973780063863353, −2.42696057016447077656324968017, −2.33453867960443266575672794357, −1.63321595585907993392196370159, −1.62667811992059918095319748907, −1.35467041291886801359594419385, −1.26929525429130014977627004030,
1.26929525429130014977627004030, 1.35467041291886801359594419385, 1.62667811992059918095319748907, 1.63321595585907993392196370159, 2.33453867960443266575672794357, 2.42696057016447077656324968017, 3.34258169756407973780063863353, 3.48743510272140338323432857979, 3.62986753811717619225181640535, 3.66302559260902772792797634801, 4.41367792985774835793984553484, 4.53966820073703962365294411420, 4.74707649054486146018078675739, 4.81688616114965219914484008719, 4.98054999661851797107399223761, 5.54181899346438456754172192226, 5.55106444768791377223910327914, 5.91446712471854276113862244309, 5.95721728587460759671213193035, 6.04205954952820237389470179231, 6.99743636863288176826705985370, 7.17774897889637640638201346750, 7.19858437646776198408952664174, 7.54874302374735635294936186923, 7.64357488615353333304081278006