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: |
3
|
Selberg data: |
(6, 33⋅56⋅113, ( :1/2,1/2,1/2), −1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
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 |
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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
−9.667750402935049765515411567581, −9.255685963136796812018701128104, −9.083657907431766356766631054875, −8.958755080134150296825081630491, −8.214905430638894958424429450593, −8.152398093026085170495732770405, −8.068013884955445269719119145717, −7.39033607072128127597643024710, −7.27222957457784295880375963242, −6.66540924706850267965456337882, −6.41431558191273704550629764140, −6.28561739412942160519080176542, −6.28062773744403945913256246318, −5.88651321407098908231315876578, −5.24767170607294390761269817532, −5.09996759719112183530139638036, −4.64017506545693089220384394903, −4.38568330778879800444946357376, −4.24391503015009162106945243657, −3.61582496401322644909572348923, −3.41778385349178823269438616811, −2.97910414254756085509959498399, −2.22506912740385525617645534196, −1.63690961981755026140006473248, −1.36951501118679666259017721234, 0, 0, 0,
1.36951501118679666259017721234, 1.63690961981755026140006473248, 2.22506912740385525617645534196, 2.97910414254756085509959498399, 3.41778385349178823269438616811, 3.61582496401322644909572348923, 4.24391503015009162106945243657, 4.38568330778879800444946357376, 4.64017506545693089220384394903, 5.09996759719112183530139638036, 5.24767170607294390761269817532, 5.88651321407098908231315876578, 6.28062773744403945913256246318, 6.28561739412942160519080176542, 6.41431558191273704550629764140, 6.66540924706850267965456337882, 7.27222957457784295880375963242, 7.39033607072128127597643024710, 8.068013884955445269719119145717, 8.152398093026085170495732770405, 8.214905430638894958424429450593, 8.958755080134150296825081630491, 9.083657907431766356766631054875, 9.255685963136796812018701128104, 9.667750402935049765515411567581