L(s) = 1 | − 4-s − 3·16-s + 2·17-s + 8·23-s + 7·25-s − 2·29-s − 10·43-s + 15·49-s − 22·53-s + 32·61-s + 3·64-s − 2·68-s + 30·79-s − 8·92-s − 7·100-s − 22·101-s + 22·103-s + 18·113-s + 2·116-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 3/4·16-s + 0.485·17-s + 1.66·23-s + 7/5·25-s − 0.371·29-s − 1.52·43-s + 15/7·49-s − 3.02·53-s + 4.09·61-s + 3/8·64-s − 0.242·68-s + 3.37·79-s − 0.834·92-s − 0.699·100-s − 2.18·101-s + 2.16·103-s + 1.69·113-s + 0.185·116-s + 3.27·121-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 + ⋯ |
Λ(s)=(=((38⋅138)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((38⋅138)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
38⋅138
|
Sign: |
1
|
Analytic conductor: |
21758.3 |
Root analytic conductor: |
3.48500 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 38⋅138, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.743394382 |
L(21) |
≈ |
3.743394382 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 13 | | 1 |
good | 2 | D4×C2 | 1+T2+p2T4+p2T6+p4T8 |
| 5 | C23 | 1−7T2+24T4−7p2T6+p4T8 |
| 7 | D4×C2 | 1−15T2+116T4−15p2T6+p4T8 |
| 11 | C22 | (1−18T2+p2T4)2 |
| 17 | D4 | (1−T+30T2−pT3+p2T4)2 |
| 19 | D4×C2 | 1−24T2+254T4−24p2T6+p4T8 |
| 23 | C2 | (1−2T+pT2)4 |
| 29 | D4 | (1+T+20T2+pT3+p2T4)2 |
| 31 | D4×C2 | 1−115T2+5224T4−115p2T6+p4T8 |
| 37 | D4×C2 | 1−79T2+3784T4−79p2T6+p4T8 |
| 41 | D4×C2 | 1−155T2+9364T4−155p2T6+p4T8 |
| 43 | D4 | (1+5T+88T2+5pT3+p2T4)2 |
| 47 | C22 | (1−26T2+p2T4)2 |
| 53 | D4 | (1+11T+98T2+11pT3+p2T4)2 |
| 59 | D4×C2 | 1−104T2+6334T4−104p2T6+p4T8 |
| 61 | D4 | (1−16T+169T2−16pT3+p2T4)2 |
| 67 | D4×C2 | 1−247T2+24124T4−247p2T6+p4T8 |
| 71 | C22 | (1+54T2+p2T4)2 |
| 73 | D4×C2 | 1−186T2+16859T4−186p2T6+p4T8 |
| 79 | D4 | (1−15T+210T2−15pT3+p2T4)2 |
| 83 | D4×C2 | 1−248T2+27454T4−248p2T6+p4T8 |
| 89 | D4×C2 | 1−160T2+16734T4−160p2T6+p4T8 |
| 97 | D4×C2 | 1−295T2+39856T4−295p2T6+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
−6.73343977855231808573855731968, −6.60135736738156978987578188732, −6.53188151316221986926612149109, −6.06467410282436298617906518941, −5.94040065779220960938064780560, −5.43106179969596408526065099035, −5.39244292840511893873551545974, −5.32425178048999822647102106585, −4.87111174822761321620364046867, −4.84936111500598726048211830937, −4.67052590880792840773830734182, −4.34974275116811973382139400763, −3.98139402152287830611797281540, −3.83474350001262017664087974199, −3.62048825687351226800046125626, −3.20798880525452539633030946132, −3.09301942385086955850903871728, −2.86683408246551183046732003024, −2.56165624439918345731330092105, −2.02155372941901206529855282017, −2.00553202849314730377912392907, −1.64666864131275140311928411999, −1.01863710236056831658686766935, −0.73891486913757544781750861954, −0.52022721083876749509999785499,
0.52022721083876749509999785499, 0.73891486913757544781750861954, 1.01863710236056831658686766935, 1.64666864131275140311928411999, 2.00553202849314730377912392907, 2.02155372941901206529855282017, 2.56165624439918345731330092105, 2.86683408246551183046732003024, 3.09301942385086955850903871728, 3.20798880525452539633030946132, 3.62048825687351226800046125626, 3.83474350001262017664087974199, 3.98139402152287830611797281540, 4.34974275116811973382139400763, 4.67052590880792840773830734182, 4.84936111500598726048211830937, 4.87111174822761321620364046867, 5.32425178048999822647102106585, 5.39244292840511893873551545974, 5.43106179969596408526065099035, 5.94040065779220960938064780560, 6.06467410282436298617906518941, 6.53188151316221986926612149109, 6.60135736738156978987578188732, 6.73343977855231808573855731968