L(s) = 1 | − 2·9-s + 8·13-s + 12·17-s + 18·25-s + 18·49-s − 24·53-s − 15·81-s − 48·101-s − 24·113-s − 16·117-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 2/3·9-s + 2.21·13-s + 2.91·17-s + 18/5·25-s + 18/7·49-s − 3.29·53-s − 5/3·81-s − 4.77·101-s − 2.25·113-s − 1.47·117-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
Λ(s)=(=((220⋅134)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((220⋅134)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
220⋅134
|
Sign: |
1
|
Analytic conductor: |
121.753 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 220⋅134, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.275294071 |
L(21) |
≈ |
3.275294071 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 13 | C2 | (1−4T+pT2)2 |
good | 3 | C22 | (1+T2+p2T4)2 |
| 5 | C22 | (1−9T2+p2T4)2 |
| 7 | C22 | (1−9T2+p2T4)2 |
| 11 | C2 | (1−pT2)4 |
| 17 | C2 | (1−3T+pT2)4 |
| 19 | C22 | (1−18T2+p2T4)2 |
| 23 | C22 | (1+26T2+p2T4)2 |
| 29 | C2 | (1+pT2)4 |
| 31 | C22 | (1+18T2+p2T4)2 |
| 37 | C22 | (1−65T2+p2T4)2 |
| 41 | C2 | (1−10T+pT2)2(1+10T+pT2)2 |
| 43 | C22 | (1+41T2+p2T4)2 |
| 47 | C22 | (1−49T2+p2T4)2 |
| 53 | C2 | (1+6T+pT2)4 |
| 59 | C22 | (1+62T2+p2T4)2 |
| 61 | C2 | (1+pT2)4 |
| 67 | C22 | (1−54T2+p2T4)2 |
| 71 | C22 | (1−97T2+p2T4)2 |
| 73 | C2 | (1−16T+pT2)2(1+16T+pT2)2 |
| 79 | C22 | (1−22T2+p2T4)2 |
| 83 | C22 | (1+14T2+p2T4)2 |
| 89 | C22 | (1+18T2+p2T4)2 |
| 97 | C22 | (1−50T2+p2T4)2 |
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
−8.052819617094294380160781551358, −8.001500452478332955298605761066, −7.78432139796203097999391745948, −7.27268070954437664544399199482, −7.23434626904243823269085128066, −6.73888529220967120930714143886, −6.59999908147912859376864130892, −6.54904934030621723450562342239, −6.05229397049117612180975765342, −5.62967365083753600460579439536, −5.61202973468214076599314075972, −5.50724896718677580597447627175, −5.21898937662854559738389325959, −4.59100940752095021947675651536, −4.57791071373344686202891073234, −4.11256054277083003311805819273, −3.86899428235325612345899844575, −3.35523302780359735111402028897, −3.10231851921943527992165824507, −3.00929405415632434968429946521, −2.87182939557841694659686341312, −2.07637111531916380067842848935, −1.41446769528959812138942316526, −1.20280345372071301713816249436, −0.840251519037930368833641877088,
0.840251519037930368833641877088, 1.20280345372071301713816249436, 1.41446769528959812138942316526, 2.07637111531916380067842848935, 2.87182939557841694659686341312, 3.00929405415632434968429946521, 3.10231851921943527992165824507, 3.35523302780359735111402028897, 3.86899428235325612345899844575, 4.11256054277083003311805819273, 4.57791071373344686202891073234, 4.59100940752095021947675651536, 5.21898937662854559738389325959, 5.50724896718677580597447627175, 5.61202973468214076599314075972, 5.62967365083753600460579439536, 6.05229397049117612180975765342, 6.54904934030621723450562342239, 6.59999908147912859376864130892, 6.73888529220967120930714143886, 7.23434626904243823269085128066, 7.27268070954437664544399199482, 7.78432139796203097999391745948, 8.001500452478332955298605761066, 8.052819617094294380160781551358