L(s) = 1 | − 8·19-s − 10·25-s + 16·31-s + 28·49-s − 4·61-s − 32·79-s − 28·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | − 1.83·19-s − 2·25-s + 2.87·31-s + 4·49-s − 0.512·61-s − 3.60·79-s − 2.68·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((216⋅312⋅54)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅312⋅54
|
Sign: |
1
|
Analytic conductor: |
88495.9 |
Root analytic conductor: |
4.15303 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.7752095040 |
L(21) |
≈ |
0.7752095040 |
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+pT2)2 |
good | 7 | C2 | (1−pT2)4 |
| 11 | C2 | (1+pT2)4 |
| 13 | C2 | (1−pT2)4 |
| 17 | C23 | 1+14T2−93T4+14p2T6+p4T8 |
| 19 | C22 | (1+4T−3T2+4pT3+p2T4)2 |
| 23 | C23 | 1−34T2+627T4−34p2T6+p4T8 |
| 29 | C2 | (1+pT2)4 |
| 31 | C22 | (1−8T+33T2−8pT3+p2T4)2 |
| 37 | C2 | (1−pT2)4 |
| 41 | C2 | (1+pT2)4 |
| 43 | C2 | (1−pT2)4 |
| 47 | C22 | (1−14T2+p2T4)2 |
| 53 | C23 | 1+86T2+4587T4+86p2T6+p4T8 |
| 59 | C2 | (1+pT2)4 |
| 61 | C22 | (1+2T−57T2+2pT3+p2T4)2 |
| 67 | C2 | (1−pT2)4 |
| 71 | C2 | (1+pT2)4 |
| 73 | C2 | (1−pT2)4 |
| 79 | C22 | (1+16T+177T2+16pT3+p2T4)2 |
| 83 | C23 | 1−154T2+16827T4−154p2T6+p4T8 |
| 89 | C2 | (1+pT2)4 |
| 97 | C2 | (1−pT2)4 |
show more | | |
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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
−6.38202947985952455331813412350, −6.24338527266760267319667286264, −6.04585073767139421091738373249, −5.75694010282644296356819880689, −5.73791444505866388055019523154, −5.46968143340456393782470328339, −5.19358487080729097839251856381, −5.05242441455579779834229351709, −4.59411762949536955101042818879, −4.45646348502415518030999229896, −4.21989337670795579039681759743, −4.15016516307278174501134584781, −3.95817035018831696407019637069, −3.75918759375632040845616557223, −3.44457491308628533649228386600, −2.92925564378062650246976573786, −2.68991472823303391582786612582, −2.63303158490606636621750028848, −2.58450369575947933647075323165, −1.97855691606459226176201303408, −1.79989341853680827774497589699, −1.55754823870179099299157319695, −1.00891017164245261018619872863, −0.807614761992021626550787113733, −0.16227331582877620967493007376,
0.16227331582877620967493007376, 0.807614761992021626550787113733, 1.00891017164245261018619872863, 1.55754823870179099299157319695, 1.79989341853680827774497589699, 1.97855691606459226176201303408, 2.58450369575947933647075323165, 2.63303158490606636621750028848, 2.68991472823303391582786612582, 2.92925564378062650246976573786, 3.44457491308628533649228386600, 3.75918759375632040845616557223, 3.95817035018831696407019637069, 4.15016516307278174501134584781, 4.21989337670795579039681759743, 4.45646348502415518030999229896, 4.59411762949536955101042818879, 5.05242441455579779834229351709, 5.19358487080729097839251856381, 5.46968143340456393782470328339, 5.73791444505866388055019523154, 5.75694010282644296356819880689, 6.04585073767139421091738373249, 6.24338527266760267319667286264, 6.38202947985952455331813412350