L(s) = 1 | + 3·2-s + 2·3-s + 6·4-s + 6·6-s + 2·7-s + 10·8-s + 4·9-s + 2·11-s + 12·12-s − 2·13-s + 6·14-s + 15·16-s − 4·17-s + 12·18-s − 3·19-s + 4·21-s + 6·22-s + 14·23-s + 20·24-s − 6·26-s + 27-s + 12·28-s + 14·29-s − 4·31-s + 21·32-s + 4·33-s − 12·34-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.15·3-s + 3·4-s + 2.44·6-s + 0.755·7-s + 3.53·8-s + 4/3·9-s + 0.603·11-s + 3.46·12-s − 0.554·13-s + 1.60·14-s + 15/4·16-s − 0.970·17-s + 2.82·18-s − 0.688·19-s + 0.872·21-s + 1.27·22-s + 2.91·23-s + 4.08·24-s − 1.17·26-s + 0.192·27-s + 2.26·28-s + 2.59·29-s − 0.718·31-s + 3.71·32-s + 0.696·33-s − 2.05·34-s + ⋯ |
Λ(s)=(=((23⋅56⋅193)s/2ΓC(s)3L(s)Λ(2−s)
Λ(s)=(=((23⋅56⋅193)s/2ΓC(s+1/2)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
23⋅56⋅193
|
Sign: |
1
|
Analytic conductor: |
436.517 |
Root analytic conductor: |
2.75423 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 23⋅56⋅193, ( :1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
22.84444621 |
L(21) |
≈ |
22.84444621 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−T)3 |
| 5 | | 1 |
| 19 | C1 | (1+T)3 |
good | 3 | S4×C2 | 1−2T+7T3−2p2T5+p3T6 |
| 7 | S4×C2 | 1−2T+3pT3−2p2T5+p3T6 |
| 11 | S4×C2 | 1−2T+T2−20T3+pT4−2p2T5+p3T6 |
| 13 | S4×C2 | 1+2T+16T2+73T3+16pT4+2p2T5+p3T6 |
| 17 | S4×C2 | 1+4T+36T2+143T3+36pT4+4p2T5+p3T6 |
| 23 | S4×C2 | 1−14T+126T2−707T3+126pT4−14p2T5+p3T6 |
| 29 | S4×C2 | 1−14T+128T2−837T3+128pT4−14p2T5+p3T6 |
| 31 | S4×C2 | 1+4T+57T2+96T3+57pT4+4p2T5+p3T6 |
| 37 | S4×C2 | 1+17T+174T2+1249T3+174pT4+17p2T5+p3T6 |
| 41 | S4×C2 | 1+8T+75T2+592T3+75pT4+8p2T5+p3T6 |
| 43 | S4×C2 | 1−2T+89T2−244T3+89pT4−2p2T5+p3T6 |
| 47 | S4×C2 | 1−13T+116T2−697T3+116pT4−13p2T5+p3T6 |
| 53 | S4×C2 | 1+10T+170T2+1057T3+170pT4+10p2T5+p3T6 |
| 59 | S4×C2 | 1+6T+148T2+533T3+148pT4+6p2T5+p3T6 |
| 61 | S4×C2 | 1−22T+255T2−2148T3+255pT4−22p2T5+p3T6 |
| 67 | S4×C2 | 1+70T2−7pT3+70pT4+p3T6 |
| 71 | S4×C2 | 1+2T+197T2+260T3+197pT4+2p2T5+p3T6 |
| 73 | S4×C2 | 1+12T+192T2+1509T3+192pT4+12p2T5+p3T6 |
| 79 | S4×C2 | 1−24T+349T2−3472T3+349pT4−24p2T5+p3T6 |
| 83 | S4×C2 | 1−12T+85T2−448T3+85pT4−12p2T5+p3T6 |
| 89 | C2 | (1+10T+pT2)3 |
| 97 | S4×C2 | 1+111T2+648T3+111pT4+p3T6 |
show more | | |
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L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.712602781636527498136243445687, −8.626159814920393922514024077340, −8.496889024136592306280115373498, −8.075894010530024812148858185685, −7.65862334787671301641646781821, −7.28025840620011693446166694910, −7.11204516473340676961782901409, −6.78734593420353788327942476600, −6.73853777901954683676849324847, −6.53049179238004494768702899980, −5.89772105921229902687536822883, −5.44986835051929002683014338050, −5.33188401267888014535227763496, −4.89242873982978485255593790771, −4.76445659554234147987757244775, −4.31480640408567750484737837572, −4.23652763277160090553895259888, −3.65140799252629995017406128079, −3.49589122725442479360417953773, −3.05165975502807886761171049816, −2.59022385295220915040614234210, −2.51140339553885340497874043869, −1.79621579347191580862917911994, −1.63930587319736415565967290568, −0.987570335650326101693547283034,
0.987570335650326101693547283034, 1.63930587319736415565967290568, 1.79621579347191580862917911994, 2.51140339553885340497874043869, 2.59022385295220915040614234210, 3.05165975502807886761171049816, 3.49589122725442479360417953773, 3.65140799252629995017406128079, 4.23652763277160090553895259888, 4.31480640408567750484737837572, 4.76445659554234147987757244775, 4.89242873982978485255593790771, 5.33188401267888014535227763496, 5.44986835051929002683014338050, 5.89772105921229902687536822883, 6.53049179238004494768702899980, 6.73853777901954683676849324847, 6.78734593420353788327942476600, 7.11204516473340676961782901409, 7.28025840620011693446166694910, 7.65862334787671301641646781821, 8.075894010530024812148858185685, 8.496889024136592306280115373498, 8.626159814920393922514024077340, 8.712602781636527498136243445687