L(s) = 1 | − 2·3-s − 4·5-s − 4·7-s + 2·9-s + 8·13-s + 8·15-s − 8·17-s − 8·19-s + 8·21-s + 2·25-s − 6·27-s + 16·35-s − 24·37-s − 16·39-s − 8·45-s − 8·49-s + 16·51-s + 16·57-s − 8·63-s − 32·65-s − 4·75-s + 11·81-s + 12·83-s + 32·85-s − 32·91-s + 32·95-s + 32·101-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s − 1.51·7-s + 2/3·9-s + 2.21·13-s + 2.06·15-s − 1.94·17-s − 1.83·19-s + 1.74·21-s + 2/5·25-s − 1.15·27-s + 2.70·35-s − 3.94·37-s − 2.56·39-s − 1.19·45-s − 8/7·49-s + 2.24·51-s + 2.11·57-s − 1.00·63-s − 3.96·65-s − 0.461·75-s + 11/9·81-s + 1.31·83-s + 3.47·85-s − 3.35·91-s + 3.28·95-s + 3.18·101-s + ⋯ |
Λ(s)=(=((228⋅34⋅54)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((228⋅34⋅54)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
228⋅34⋅54
|
Sign: |
1
|
Analytic conductor: |
55247.5 |
Root analytic conductor: |
3.91551 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 228⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.07562180172 |
L(21) |
≈ |
0.07562180172 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C22 | 1+2T+2T2+2pT3+p2T4 |
| 5 | C2 | (1+2T+pT2)2 |
good | 7 | D4 | (1+2T+10T2+2pT3+p2T4)2 |
| 11 | C22 | (1−6T2+p2T4)2 |
| 13 | D4 | (1−4T+10T2−4pT3+p2T4)2 |
| 17 | D4 | (1+4T+18T2+4pT3+p2T4)2 |
| 19 | D4 | (1+4T+22T2+4pT3+p2T4)2 |
| 23 | D4×C2 | 1−80T2+2638T4−80p2T6+p4T8 |
| 29 | C22 | (1+38T2+p2T4)2 |
| 31 | C4×C2 | 1−76T2+3046T4−76p2T6+p4T8 |
| 37 | D4 | (1+12T+90T2+12pT3+p2T4)2 |
| 41 | C22 | (1−66T2+p2T4)2 |
| 43 | D4×C2 | 1−64T2+3742T4−64p2T6+p4T8 |
| 47 | D4×C2 | 1−80T2+5038T4−80p2T6+p4T8 |
| 53 | C22 | (1−26T2+p2T4)2 |
| 59 | C22 | (1−38T2+p2T4)2 |
| 61 | D4×C2 | 1−52T2+2998T4−52p2T6+p4T8 |
| 67 | D4×C2 | 1−128T2+8574T4−128p2T6+p4T8 |
| 71 | C2 | (1+pT2)4 |
| 73 | D4×C2 | 1−100T2+8038T4−100p2T6+p4T8 |
| 79 | D4×C2 | 1−204T2+20006T4−204p2T6+p4T8 |
| 83 | D4 | (1−6T+50T2−6pT3+p2T4)2 |
| 89 | C22 | (1−114T2+p2T4)2 |
| 97 | D4×C2 | 1−196T2+23302T4−196p2T6+p4T8 |
show more | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.52211982496569215610738779668, −6.50405224968986190626471867064, −6.37568268827427035038902799427, −5.80660121952352394802535089679, −5.80213177917306486313438171353, −5.56493272807924006166457090284, −5.43821716337003280411871969304, −4.90076460901012061773843165093, −4.75621365085638684658952130187, −4.62855873528324580411610889894, −4.37142055357000575672811179246, −4.13271124916776380689088033413, −3.77822380228752773342998721276, −3.68038995722779402458139277682, −3.53417922306261054969677734272, −3.42624119283966729587300551481, −3.19993331502472225196442097619, −2.66806962355898017103246054473, −2.33119268148321605330572802028, −2.05576500429050122653053643896, −1.65209907119912943721267385622, −1.62477496664293442295210721671, −0.956765282153551266667044222471, −0.32758658200646278590944924021, −0.14537017013971243800965563307,
0.14537017013971243800965563307, 0.32758658200646278590944924021, 0.956765282153551266667044222471, 1.62477496664293442295210721671, 1.65209907119912943721267385622, 2.05576500429050122653053643896, 2.33119268148321605330572802028, 2.66806962355898017103246054473, 3.19993331502472225196442097619, 3.42624119283966729587300551481, 3.53417922306261054969677734272, 3.68038995722779402458139277682, 3.77822380228752773342998721276, 4.13271124916776380689088033413, 4.37142055357000575672811179246, 4.62855873528324580411610889894, 4.75621365085638684658952130187, 4.90076460901012061773843165093, 5.43821716337003280411871969304, 5.56493272807924006166457090284, 5.80213177917306486313438171353, 5.80660121952352394802535089679, 6.37568268827427035038902799427, 6.50405224968986190626471867064, 6.52211982496569215610738779668