L(s) = 1 | − 5·2-s − 537·4-s + 2.50e3·5-s − 9.60e3·7-s + 991·8-s − 1.25e4·10-s − 6.45e4·11-s − 2.93e4·13-s + 4.80e4·14-s + 1.69e5·16-s − 2.78e5·17-s − 9.29e5·19-s − 1.34e6·20-s + 3.22e5·22-s + 5.26e5·23-s + 3.90e6·25-s + 1.46e5·26-s + 5.15e6·28-s − 3.65e5·29-s − 4.97e6·31-s − 5.47e5·32-s + 1.39e6·34-s − 2.40e7·35-s + 2.28e7·37-s + 4.64e6·38-s + 2.47e6·40-s + 2.82e7·41-s + ⋯ |
L(s) = 1 | − 0.220·2-s − 1.04·4-s + 1.78·5-s − 1.51·7-s + 0.0855·8-s − 0.395·10-s − 1.32·11-s − 0.285·13-s + 0.334·14-s + 0.648·16-s − 0.809·17-s − 1.63·19-s − 1.87·20-s + 0.293·22-s + 0.391·23-s + 2·25-s + 0.0630·26-s + 1.58·28-s − 0.0960·29-s − 0.968·31-s − 0.0922·32-s + 0.178·34-s − 2.70·35-s + 2.00·37-s + 0.361·38-s + 0.153·40-s + 1.56·41-s + ⋯ |
Λ(s)=(=((38⋅54⋅74)s/2ΓC(s)4L(s)Λ(10−s)
Λ(s)=(=((38⋅54⋅74)s/2ΓC(s+9/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
38⋅54⋅74
|
Sign: |
1
|
Analytic conductor: |
6.92774×108 |
Root analytic conductor: |
12.7372 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 38⋅54⋅74, ( :9/2,9/2,9/2,9/2), 1)
|
Particular Values
L(5) |
= |
0 |
L(21) |
= |
0 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.32286952252166934542356455881, −7.19455851909712716526031228806, −6.78538779704046950693374848133, −6.53272801875708598740682084733, −6.22881664407345681420619445058, −6.10960683413165327726538418903, −5.83007043394838359157805641403, −5.66810737170310285334505618964, −5.64228315898845193410868676009, −4.87290866606344729493325032800, −4.78215578802665757611940850140, −4.65058392081981738471723867619, −4.51572843527356287686198526505, −3.84105730711281393251247700434, −3.71970605952549733346946210914, −3.57487286213784815634382578955, −2.96229214630390096444131852386, −2.66836263546744812821894502860, −2.57363592073880507451737530294, −2.34714115200587533640154134117, −2.16340944345734252717458307741, −1.76216374968790785481751072595, −1.13954627883648156359915491816, −1.00762893299744498564416093512, −0.959118277880721762417210956129, 0, 0, 0, 0,
0.959118277880721762417210956129, 1.00762893299744498564416093512, 1.13954627883648156359915491816, 1.76216374968790785481751072595, 2.16340944345734252717458307741, 2.34714115200587533640154134117, 2.57363592073880507451737530294, 2.66836263546744812821894502860, 2.96229214630390096444131852386, 3.57487286213784815634382578955, 3.71970605952549733346946210914, 3.84105730711281393251247700434, 4.51572843527356287686198526505, 4.65058392081981738471723867619, 4.78215578802665757611940850140, 4.87290866606344729493325032800, 5.64228315898845193410868676009, 5.66810737170310285334505618964, 5.83007043394838359157805641403, 6.10960683413165327726538418903, 6.22881664407345681420619445058, 6.53272801875708598740682084733, 6.78538779704046950693374848133, 7.19455851909712716526031228806, 7.32286952252166934542356455881