L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s − 17-s + 19-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 37-s − 39-s − 41-s − 43-s − 45-s + 47-s − 51-s + 53-s + 55-s + 57-s + 59-s − 61-s + 65-s + ⋯ |
L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s − 17-s + 19-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 37-s − 39-s − 41-s − 43-s − 45-s + 47-s − 51-s + 53-s + 55-s + 57-s + 59-s − 61-s + 65-s + ⋯ |
Λ(s)=(=(28s/2ΓR(s)L(s)Λ(1−s)
Λ(s)=(=(28s/2ΓR(s)L(s)Λ(1−s)
Degree: |
1 |
Conductor: |
28
= 22⋅7
|
Sign: |
1
|
Analytic conductor: |
0.130031 |
Root analytic conductor: |
0.130031 |
Motivic weight: |
0 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
χ28(27,⋅)
|
Primitive: |
yes
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(1, 28, (0: ), 1)
|
Particular Values
L(21) |
≈ |
0.8223610378 |
L(21) |
≈ |
0.8223610378 |
L(1) |
≈ |
1.046454884 |
L(1) |
≈ |
1.046454884 |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+T |
| 5 | 1−T |
| 11 | 1−T |
| 13 | 1−T |
| 17 | 1−T |
| 19 | 1+T |
| 23 | 1−T |
| 29 | 1+T |
| 31 | 1+T |
| 37 | 1+T |
| 41 | 1−T |
| 43 | 1−T |
| 47 | 1+T |
| 53 | 1+T |
| 59 | 1+T |
| 61 | 1−T |
| 67 | 1−T |
| 71 | 1−T |
| 73 | 1−T |
| 79 | 1−T |
| 83 | 1+T |
| 89 | 1−T |
| 97 | 1−T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−37.512179947842773084657930490390, −36.36045602124227699396162943688, −35.268334030341376930207529526020, −33.849591836732886972765039184334, −32.184175727883333033755618747177, −31.36586297616383901440906943622, −30.39449098988413064026229502805, −28.72073200609003118776449308629, −27.02957442259405572955262877191, −26.38179066809542195016806658376, −24.73886680489813285047033853957, −23.691534626557196405383352007920, −21.9888981476056831334669883160, −20.39328285450433247152743984794, −19.538616099899411451627397917207, −18.20042527549262445931043231016, −16.01626147715059280892693066920, −15.06829496191904340653215529897, −13.544286371829787414922340082154, −12.02206203818595380135277821042, −10.116306854731297791099024344636, −8.39512407283755124263059308174, −7.29088506700227937478527206798, −4.526774087015476109286965871473, −2.77728324207659233094209284012,
2.77728324207659233094209284012, 4.526774087015476109286965871473, 7.29088506700227937478527206798, 8.39512407283755124263059308174, 10.116306854731297791099024344636, 12.02206203818595380135277821042, 13.544286371829787414922340082154, 15.06829496191904340653215529897, 16.01626147715059280892693066920, 18.20042527549262445931043231016, 19.538616099899411451627397917207, 20.39328285450433247152743984794, 21.9888981476056831334669883160, 23.691534626557196405383352007920, 24.73886680489813285047033853957, 26.38179066809542195016806658376, 27.02957442259405572955262877191, 28.72073200609003118776449308629, 30.39449098988413064026229502805, 31.36586297616383901440906943622, 32.184175727883333033755618747177, 33.849591836732886972765039184334, 35.268334030341376930207529526020, 36.36045602124227699396162943688, 37.512179947842773084657930490390