L(s) = 1 | − 6·3-s + 16·9-s + 4·13-s − 6·17-s − 6·19-s + 6·23-s − 14·25-s − 24·27-s + 12·29-s − 8·37-s − 24·39-s − 18·41-s − 36·43-s − 10·49-s + 36·51-s + 36·57-s − 12·59-s + 36·61-s − 6·67-s − 36·69-s − 30·71-s + 84·75-s + 24·79-s + 21·81-s − 24·83-s − 72·87-s − 12·89-s + ⋯ |
L(s) = 1 | − 3.46·3-s + 16/3·9-s + 1.10·13-s − 1.45·17-s − 1.37·19-s + 1.25·23-s − 2.79·25-s − 4.61·27-s + 2.22·29-s − 1.31·37-s − 3.84·39-s − 2.81·41-s − 5.48·43-s − 1.42·49-s + 5.04·51-s + 4.76·57-s − 1.56·59-s + 4.60·61-s − 0.733·67-s − 4.33·69-s − 3.56·71-s + 9.69·75-s + 2.70·79-s + 7/3·81-s − 2.63·83-s − 7.71·87-s − 1.27·89-s + ⋯ |
Λ(s)=(=((224⋅134)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((224⋅134)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
224⋅134
|
Sign: |
1
|
Analytic conductor: |
1948.05 |
Root analytic conductor: |
2.57750 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.1405098498 |
L(21) |
≈ |
0.1405098498 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 13 | C22 | 1−4T+3T2−4pT3+p2T4 |
good | 3 | D4×C2 | 1+2pT+20T2+16pT3+91T4+16p2T5+20p2T6+2p4T7+p4T8 |
| 5 | C22 | (1+7T2+p2T4)2 |
| 7 | C23 | 1+10T2+51T4+10p2T6+p4T8 |
| 11 | C23 | 1−10T2−21T4−10p2T6+p4T8 |
| 17 | D4×C2 | 1+6T+5T2−18T3+60T4−18pT5+5p2T6+6p3T7+p4T8 |
| 19 | D4×C2 | 1+6T−8T2+36T3+891T4+36pT5−8p2T6+6p3T7+p4T8 |
| 23 | D4×C2 | 1−6T−16T2−36T3+1347T4−36pT5−16p2T6−6p3T7+p4T8 |
| 29 | D4×C2 | 1−12T+109T2−732T3+4272T4−732pT5+109p2T6−12p3T7+p4T8 |
| 31 | D4×C2 | 1−20T2−678T4−20p2T6+p4T8 |
| 37 | D4×C2 | 1+8T+T2−88T3+232T4−88pT5+p2T6+8p3T7+p4T8 |
| 41 | C22 | (1+9T+68T2+9pT3+p2T4)2 |
| 43 | D4×C2 | 1+36T+622T2+6840T3+52827T4+6840pT5+622p2T6+36p3T7+p4T8 |
| 47 | D4×C2 | 1−20T2+1818T4−20p2T6+p4T8 |
| 53 | D4×C2 | 1−26T2+1899T4−26p2T6+p4T8 |
| 59 | D4×C2 | 1+12T+38T2−144T3−741T4−144pT5+38p2T6+12p3T7+p4T8 |
| 61 | D4×C2 | 1−36T+653T2−7956T3+71472T4−7956pT5+653p2T6−36p3T7+p4T8 |
| 67 | D4×C2 | 1+6T−80T2−108T3+6555T4−108pT5−80p2T6+6p3T7+p4T8 |
| 71 | D4×C2 | 1+30T+508T2+6240T3+59523T4+6240pT5+508p2T6+30p3T7+p4T8 |
| 73 | C22 | (1−119T2+p2T4)2 |
| 79 | C2 | (1−6T+pT2)4 |
| 83 | D4 | (1+12T+190T2+12pT3+p2T4)2 |
| 89 | D4×C2 | 1+12T+94T2+552T3−1533T4+552pT5+94p2T6+12p3T7+p4T8 |
| 97 | C23 | 1+158T2+15555T4+158p2T6+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
−7.03366007459414717175505264081, −6.89996849363826723968302867436, −6.83888435743621444972540711504, −6.57360282905079723222950624988, −6.40636232791891529767911468965, −6.17560028935469063834556694163, −6.03931729273489295845264799227, −5.53209272287889459887740227687, −5.47007774894861552043292521614, −5.33961648966361732952469451127, −5.26524818147576124065907596841, −4.72668638025934901512950445731, −4.58934915258781921975665968458, −4.34557721680566296593583925808, −4.32168144835051336123744765097, −3.67942922941807242589105191151, −3.46179129825555789843229820554, −3.10090611796105007247769626061, −3.04721831298631488284570755128, −2.24988707983899543841508529829, −1.82910717526818829491881239249, −1.55733465656631249562244936218, −1.51583098123029121049522291003, −0.38385948736561622085202010658, −0.29863194188471402552217509324,
0.29863194188471402552217509324, 0.38385948736561622085202010658, 1.51583098123029121049522291003, 1.55733465656631249562244936218, 1.82910717526818829491881239249, 2.24988707983899543841508529829, 3.04721831298631488284570755128, 3.10090611796105007247769626061, 3.46179129825555789843229820554, 3.67942922941807242589105191151, 4.32168144835051336123744765097, 4.34557721680566296593583925808, 4.58934915258781921975665968458, 4.72668638025934901512950445731, 5.26524818147576124065907596841, 5.33961648966361732952469451127, 5.47007774894861552043292521614, 5.53209272287889459887740227687, 6.03931729273489295845264799227, 6.17560028935469063834556694163, 6.40636232791891529767911468965, 6.57360282905079723222950624988, 6.83888435743621444972540711504, 6.89996849363826723968302867436, 7.03366007459414717175505264081