L(s) = 1 | + 8·9-s − 10·25-s − 24·29-s + 48·41-s − 8·49-s + 30·81-s + 72·101-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 | + 8/3·9-s − 2·25-s − 4.45·29-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 7.16·101-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⋅54⋅234)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((216⋅54⋅234)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅54⋅234
|
Sign: |
1
|
Analytic conductor: |
46599.3 |
Root analytic conductor: |
3.83307 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅54⋅234, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.169088500 |
L(21) |
≈ |
3.169088500 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C2 | (1+pT2)2 |
| 23 | C22 | 1−44T2+p2T4 |
good | 3 | C22 | (1−4T2+p2T4)2 |
| 7 | C22 | (1+4T2+p2T4)2 |
| 11 | C2 | (1+pT2)4 |
| 13 | C2 | (1−pT2)4 |
| 17 | C2 | (1+pT2)4 |
| 19 | C2 | (1+pT2)4 |
| 29 | C2 | (1+6T+pT2)4 |
| 31 | C2 | (1−pT2)4 |
| 37 | C2 | (1+pT2)4 |
| 41 | C2 | (1−12T+pT2)4 |
| 43 | C22 | (1+76T2+p2T4)2 |
| 47 | C22 | (1+4T2+p2T4)2 |
| 53 | C2 | (1+pT2)4 |
| 59 | C2 | (1−pT2)4 |
| 61 | C2 | (1−8T+pT2)2(1+8T+pT2)2 |
| 67 | C22 | (1−116T2+p2T4)2 |
| 71 | C2 | (1−pT2)4 |
| 73 | C2 | (1−pT2)4 |
| 79 | C2 | (1+pT2)4 |
| 83 | C22 | (1+76T2+p2T4)2 |
| 89 | C2 | (1−6T+pT2)2(1+6T+pT2)2 |
| 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
−6.53030672977214159267916621211, −6.33830228616134470610451434397, −6.05332309495927469186098916214, −5.95054161269808339431890328114, −5.87828133552865796052554823830, −5.49144701961845764894916238254, −5.44125879953846344446454867939, −4.98576917281967103727691240206, −4.94654517481940278330848550806, −4.32511251589903727092955923506, −4.24622284453921309041975398225, −4.22353258160984330676185182317, −4.07293377587954730748701419088, −3.97095303718592121737034638264, −3.36165500083336533606413305160, −3.29590481158386921081225895181, −3.20672562077088951890911286649, −2.35789569964417640388409082685, −2.26038205265691797041900241180, −2.12864841305706741144718164091, −1.98267923644825071147193256303, −1.50523716604757432411317030971, −1.05077269191663283650414639431, −0.992719158316320777705041565287, −0.31706902330945866442491259851,
0.31706902330945866442491259851, 0.992719158316320777705041565287, 1.05077269191663283650414639431, 1.50523716604757432411317030971, 1.98267923644825071147193256303, 2.12864841305706741144718164091, 2.26038205265691797041900241180, 2.35789569964417640388409082685, 3.20672562077088951890911286649, 3.29590481158386921081225895181, 3.36165500083336533606413305160, 3.97095303718592121737034638264, 4.07293377587954730748701419088, 4.22353258160984330676185182317, 4.24622284453921309041975398225, 4.32511251589903727092955923506, 4.94654517481940278330848550806, 4.98576917281967103727691240206, 5.44125879953846344446454867939, 5.49144701961845764894916238254, 5.87828133552865796052554823830, 5.95054161269808339431890328114, 6.05332309495927469186098916214, 6.33830228616134470610451434397, 6.53030672977214159267916621211