L(s) = 1 | + 2·3-s − 5-s + 7·7-s − 9-s + 3·13-s − 2·15-s − 4·17-s + 8·19-s + 14·21-s + 9·23-s − 9·25-s − 7·27-s − 9·29-s + 17·31-s − 7·35-s − 3·37-s + 6·39-s − 16·41-s − 4·43-s + 45-s + 11·47-s + 18·49-s − 8·51-s − 3·53-s + 16·57-s − 2·59-s + 15·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s + 2.64·7-s − 1/3·9-s + 0.832·13-s − 0.516·15-s − 0.970·17-s + 1.83·19-s + 3.05·21-s + 1.87·23-s − 9/5·25-s − 1.34·27-s − 1.67·29-s + 3.05·31-s − 1.18·35-s − 0.493·37-s + 0.960·39-s − 2.49·41-s − 0.609·43-s + 0.149·45-s + 1.60·47-s + 18/7·49-s − 1.12·51-s − 0.412·53-s + 2.11·57-s − 0.260·59-s + 1.92·61-s + ⋯ |
Λ(s)=(=(25934336s/2ΓC(s)3L(s)Λ(2−s)
Λ(s)=(=(25934336s/2ΓC(s+1/2)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
25934336
= 29⋅373
|
Sign: |
1
|
Analytic conductor: |
13.2040 |
Root analytic conductor: |
1.53739 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 25934336, ( :1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.258191018 |
L(21) |
≈ |
3.258191018 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 37 | C1 | (1+T)3 |
good | 3 | S4×C2 | 1−2T+5T2−5T3+5pT4−2p2T5+p3T6 |
| 5 | S4×C2 | 1+T+2pT2+12T3+2p2T4+p2T5+p3T6 |
| 7 | S4×C2 | 1−pT+31T2−94T3+31pT4−p3T5+p3T6 |
| 11 | S4×C2 | 1−3T2+27T3−3pT4+p3T6 |
| 13 | S4×C2 | 1−3T+6T2−16T3+6pT4−3p2T5+p3T6 |
| 17 | S4×C2 | 1+4T+31T2+120T3+31pT4+4p2T5+p3T6 |
| 19 | S4×C2 | 1−8T+53T2−240T3+53pT4−8p2T5+p3T6 |
| 23 | S4×C2 | 1−9T+4pT2−428T3+4p2T4−9p2T5+p3T6 |
| 29 | S4×C2 | 1+9T+110T2+536T3+110pT4+9p2T5+p3T6 |
| 31 | S4×C2 | 1−17T+184T2−1202T3+184pT4−17p2T5+p3T6 |
| 41 | S4×C2 | 1+16T+193T2+1359T3+193pT4+16p2T5+p3T6 |
| 43 | S4×C2 | 1+4T+9T2+112T3+9pT4+4p2T5+p3T6 |
| 47 | S4×C2 | 1−11T+131T2−1030T3+131pT4−11p2T5+p3T6 |
| 53 | S4×C2 | 1+3T+59T2+610T3+59pT4+3p2T5+p3T6 |
| 59 | S4×C2 | 1+2T+53T2+220T3+53pT4+2p2T5+p3T6 |
| 61 | S4×C2 | 1−15T+212T2−1778T3+212pT4−15p2T5+p3T6 |
| 67 | S4×C2 | 1+5T+22T2−274T3+22pT4+5p2T5+p3T6 |
| 71 | S4×C2 | 1+5T+189T2+714T3+189pT4+5p2T5+p3T6 |
| 73 | S4×C2 | 1+6T+195T2+839T3+195pT4+6p2T5+p3T6 |
| 79 | S4×C2 | 1+T+218T2+126T3+218pT4+p2T5+p3T6 |
| 83 | S4×C2 | 1+9T+173T2+1606T3+173pT4+9p2T5+p3T6 |
| 89 | S4×C2 | 1−16T+3pT2−2784T3+3p2T4−16p2T5+p3T6 |
| 97 | S4×C2 | 1+47T2+256T3+47pT4+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
−10.50667674794395328904484860416, −10.24559456423596935575688897165, −9.844634532495756245168498798666, −9.547408148510392708352804041462, −8.869996196977424984266331586994, −8.863899326492502878201953865086, −8.627342280285411058871019505602, −8.293683295342721729660434166900, −7.88637576847425579767411056500, −7.83502155766063885308161734339, −7.46566984707687402433081972410, −7.01388337253448340997932032971, −6.71769660084259130592038541006, −6.07682723621157549923626312270, −5.54008877409685999977956279070, −5.46159198185251203645560997923, −4.86530737922999155740176177980, −4.63674996738275428236758451517, −4.27208244246088495069323457097, −3.56361983012300239336936775696, −3.34243059582778935043951132583, −2.84942306405856913627278075228, −2.14416178081521092823242555565, −1.75209515526012064562747615162, −1.12037709954869759125968838726,
1.12037709954869759125968838726, 1.75209515526012064562747615162, 2.14416178081521092823242555565, 2.84942306405856913627278075228, 3.34243059582778935043951132583, 3.56361983012300239336936775696, 4.27208244246088495069323457097, 4.63674996738275428236758451517, 4.86530737922999155740176177980, 5.46159198185251203645560997923, 5.54008877409685999977956279070, 6.07682723621157549923626312270, 6.71769660084259130592038541006, 7.01388337253448340997932032971, 7.46566984707687402433081972410, 7.83502155766063885308161734339, 7.88637576847425579767411056500, 8.293683295342721729660434166900, 8.627342280285411058871019505602, 8.863899326492502878201953865086, 8.869996196977424984266331586994, 9.547408148510392708352804041462, 9.844634532495756245168498798666, 10.24559456423596935575688897165, 10.50667674794395328904484860416