L(s) = 1 | + 2-s + 3·3-s − 2·4-s + 3·6-s + 4·7-s − 2·8-s + 6·9-s + 3·11-s − 6·12-s + 2·13-s + 4·14-s + 16-s + 6·18-s − 6·19-s + 12·21-s + 3·22-s + 12·23-s − 6·24-s + 2·26-s + 10·27-s − 8·28-s + 8·29-s − 8·31-s − 32-s + 9·33-s − 12·36-s + 4·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s − 4-s + 1.22·6-s + 1.51·7-s − 0.707·8-s + 2·9-s + 0.904·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.41·18-s − 1.37·19-s + 2.61·21-s + 0.639·22-s + 2.50·23-s − 1.22·24-s + 0.392·26-s + 1.92·27-s − 1.51·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s + 1.56·33-s − 2·36-s + 0.657·37-s + ⋯ |
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s)3L(s)Λ(2−s)
Λ(s)=(=((33⋅56⋅113)s/2ΓC(s+1/2)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
33⋅56⋅113
|
Sign: |
1
|
Analytic conductor: |
285.886 |
Root analytic conductor: |
2.56664 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 33⋅56⋅113, ( :1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
8.887783193 |
L(21) |
≈ |
8.887783193 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | C1 | (1−T)3 |
| 5 | | 1 |
| 11 | C1 | (1−T)3 |
good | 2 | S4×C2 | 1−T+3T2−3T3+3pT4−p2T5+p3T6 |
| 7 | S4×C2 | 1−4T+3pT2−52T3+3p2T4−4p2T5+p3T6 |
| 13 | S4×C2 | 1−2T+35T2−48T3+35pT4−2p2T5+p3T6 |
| 17 | S4×C2 | 1+23T2+52T3+23pT4+p3T6 |
| 19 | S4×C2 | 1+6T+53T2+188T3+53pT4+6p2T5+p3T6 |
| 23 | C2 | (1−4T+pT2)3 |
| 29 | S4×C2 | 1−8T+71T2−304T3+71pT4−8p2T5+p3T6 |
| 31 | S4×C2 | 1+8T+101T2+480T3+101pT4+8p2T5+p3T6 |
| 37 | S4×C2 | 1−4T+95T2−264T3+95pT4−4p2T5+p3T6 |
| 41 | S4×C2 | 1−8T+11T2+272T3+11pT4−8p2T5+p3T6 |
| 43 | S4×C2 | 1−12T+3pT2−884T3+3p2T4−12p2T5+p3T6 |
| 47 | S4×C2 | 1−16T+189T2−1536T3+189pT4−16p2T5+p3T6 |
| 53 | S4×C2 | 1−16T+191T2−1712T3+191pT4−16p2T5+p3T6 |
| 59 | S4×C2 | 1−8T+113T2−1024T3+113pT4−8p2T5+p3T6 |
| 61 | S4×C2 | 1+22T+291T2+2692T3+291pT4+22p2T5+p3T6 |
| 67 | S4×C2 | 1−12T+185T2−1544T3+185pT4−12p2T5+p3T6 |
| 71 | S4×C2 | 1+12T+117T2+760T3+117pT4+12p2T5+p3T6 |
| 73 | S4×C2 | 1+10T+175T2+1072T3+175pT4+10p2T5+p3T6 |
| 79 | S4×C2 | 1+10T+25T2−140T3+25pT4+10p2T5+p3T6 |
| 83 | S4×C2 | 1+10T+189T2+1056T3+189pT4+10p2T5+p3T6 |
| 89 | S4×C2 | 1−18T+255T2−2684T3+255pT4−18p2T5+p3T6 |
| 97 | S4×C2 | 1−16T+131T2−672T3+131pT4−16p2T5+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
−9.104203974436054120229060692702, −8.765401354852331276024806660120, −8.614624849082227082319612573810, −8.567757934604622706684121475401, −7.87795856184446085183920192508, −7.79470942502315420280141410663, −7.41889197249961132116921462587, −7.26608016874994634906267709187, −6.74837658191249750479556754827, −6.66843397542123878391877571256, −5.97651217865648722128749755082, −5.73901189515565344498359566074, −5.52329144866696438507384362168, −4.73497171417031631695587691727, −4.70395705559365580448502452741, −4.59649657740377297783744434237, −4.16173390288034934150778132376, −3.80115792904191761314367007499, −3.74425489410673704852611651654, −2.96594741682823080045916528160, −2.79104211969081013802893288998, −2.29615279519136901163764750232, −1.87826704779462494268479947443, −1.10772739208189699857074096964, −1.05530810322569037039931007084,
1.05530810322569037039931007084, 1.10772739208189699857074096964, 1.87826704779462494268479947443, 2.29615279519136901163764750232, 2.79104211969081013802893288998, 2.96594741682823080045916528160, 3.74425489410673704852611651654, 3.80115792904191761314367007499, 4.16173390288034934150778132376, 4.59649657740377297783744434237, 4.70395705559365580448502452741, 4.73497171417031631695587691727, 5.52329144866696438507384362168, 5.73901189515565344498359566074, 5.97651217865648722128749755082, 6.66843397542123878391877571256, 6.74837658191249750479556754827, 7.26608016874994634906267709187, 7.41889197249961132116921462587, 7.79470942502315420280141410663, 7.87795856184446085183920192508, 8.567757934604622706684121475401, 8.614624849082227082319612573810, 8.765401354852331276024806660120, 9.104203974436054120229060692702