L(s) = 1 | + 3.18e6·3-s + 2.91e8·5-s − 1.21e11·7-s + 7.62e12·9-s + 2.31e14·11-s − 1.15e15·13-s + 9.29e14·15-s − 4.61e16·17-s + 2.68e17·19-s − 3.87e17·21-s − 1.29e18·23-s + 3.00e18·25-s + 1.62e19·27-s + 6.01e19·29-s − 2.03e20·31-s + 7.39e20·33-s − 3.54e19·35-s + 3.02e21·37-s − 3.68e21·39-s − 1.08e22·41-s + 1.29e21·43-s + 2.22e21·45-s + 9.93e22·47-s − 5.76e22·49-s − 1.47e23·51-s − 3.57e22·53-s + 6.75e22·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.106·5-s − 0.474·7-s + 9-s + 2.02·11-s − 1.05·13-s + 0.123·15-s − 1.12·17-s + 1.46·19-s − 0.547·21-s − 0.537·23-s + 0.402·25-s + 0.769·27-s + 1.08·29-s − 1.49·31-s + 2.33·33-s − 0.0506·35-s + 2.03·37-s − 1.22·39-s − 1.83·41-s + 0.115·43-s + 0.106·45-s + 2.65·47-s − 0.876·49-s − 1.30·51-s − 0.188·53-s + 0.216·55-s + ⋯ |
Λ(s)=(=(2304s/2ΓC(s)2L(s)Λ(28−s)
Λ(s)=(=(2304s/2ΓC(s+27/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2304
= 28⋅32
|
Sign: |
1
|
Analytic conductor: |
49146.7 |
Root analytic conductor: |
14.8892 |
Motivic weight: |
27 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 2304, ( :27/2,27/2), 1)
|
Particular Values
L(14) |
≈ |
7.609185109 |
L(21) |
≈ |
7.609185109 |
L(229) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.94436693500532443005117157072, −10.16615095017602300676912623779, −9.690470117714450293190299982649, −9.330698331600928852650419157591, −8.865676492525466583574453318746, −8.495997507737936948496271897935, −7.48621736765211734997499098291, −7.30796439538500621827271816730, −6.70625691975758057482531836782, −6.18841066958626046258365738347, −5.50476369418244116681712293780, −4.59647152603746605198860374674, −4.28148016975867483145426968632, −3.75246637682478132245336427694, −2.96148040917870694438810242911, −2.88897734265127925217414309474, −1.81038388398900894849820668871, −1.80636949032861795819643728766, −0.865440736491405725008687668011, −0.52071941934844237508838271620,
0.52071941934844237508838271620, 0.865440736491405725008687668011, 1.80636949032861795819643728766, 1.81038388398900894849820668871, 2.88897734265127925217414309474, 2.96148040917870694438810242911, 3.75246637682478132245336427694, 4.28148016975867483145426968632, 4.59647152603746605198860374674, 5.50476369418244116681712293780, 6.18841066958626046258365738347, 6.70625691975758057482531836782, 7.30796439538500621827271816730, 7.48621736765211734997499098291, 8.495997507737936948496271897935, 8.865676492525466583574453318746, 9.330698331600928852650419157591, 9.690470117714450293190299982649, 10.16615095017602300676912623779, 10.94436693500532443005117157072