L(s) = 1 | + 2-s − 3·3-s − 3·6-s + 2·8-s + 6·9-s + 3·11-s + 2·13-s + 3·16-s + 2·17-s + 6·18-s + 8·19-s + 3·22-s − 6·24-s + 2·26-s − 10·27-s − 10·29-s + 8·31-s + 3·32-s − 9·33-s + 2·34-s + 6·37-s + 8·38-s − 6·39-s − 14·41-s − 4·43-s + 8·47-s − 9·48-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s + 0.707·8-s + 2·9-s + 0.904·11-s + 0.554·13-s + 3/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 0.639·22-s − 1.22·24-s + 0.392·26-s − 1.92·27-s − 1.85·29-s + 1.43·31-s + 0.530·32-s − 1.56·33-s + 0.342·34-s + 0.986·37-s + 1.29·38-s − 0.960·39-s − 2.18·41-s − 0.609·43-s + 1.16·47-s − 1.29·48-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) |
≈ |
2.743864766 |
L(21) |
≈ |
2.743864766 |
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+T2−3T3+pT4−p2T5+p3T6 |
| 7 | S4×C2 | 1+5T2−16T3+5pT4+p3T6 |
| 13 | D6 | 1−2T+27T2−44T3+27pT4−2p2T5+p3T6 |
| 17 | S4×C2 | 1−2T−T2+116T3−pT4−2p2T5+p3T6 |
| 19 | S4×C2 | 1−8T+41T2−144T3+41pT4−8p2T5+p3T6 |
| 23 | S4×C2 | 1+5T2+128T3+5pT4+p3T6 |
| 29 | S4×C2 | 1+10T+99T2+540T3+99pT4+10p2T5+p3T6 |
| 31 | S4×C2 | 1−8T+61T2−368T3+61pT4−8p2T5+p3T6 |
| 37 | C2 | (1−2T+pT2)3 |
| 41 | S4×C2 | 1+14T+167T2+1156T3+167pT4+14p2T5+p3T6 |
| 43 | S4×C2 | 1+4T+49T2−56T3+49pT4+4p2T5+p3T6 |
| 47 | S4×C2 | 1−8T+109T2−624T3+109pT4−8p2T5+p3T6 |
| 53 | S4×C2 | 1−6T+107T2−644T3+107pT4−6p2T5+p3T6 |
| 59 | S4×C2 | 1−12T+161T2−1096T3+161pT4−12p2T5+p3T6 |
| 61 | S4×C2 | 1+6T+131T2+484T3+131pT4+6p2T5+p3T6 |
| 67 | S4×C2 | 1−4T+153T2−472T3+153pT4−4p2T5+p3T6 |
| 71 | S4×C2 | 1−8T+181T2−1008T3+181pT4−8p2T5+p3T6 |
| 73 | S4×C2 | 1−14T+223T2−1700T3+223pT4−14p2T5+p3T6 |
| 79 | S4×C2 | 1−12T+173T2−1096T3+173pT4−12p2T5+p3T6 |
| 83 | S4×C2 | 1+129T2−16T3+129pT4+p3T6 |
| 89 | S4×C2 | 1+10T+215T2+1580T3+215pT4+10p2T5+p3T6 |
| 97 | S4×C2 | 1+22T+399T2+4276T3+399pT4+22p2T5+p3T6 |
show more | | |
show less | | |
L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.509729547902967316931157586557, −8.719332441510926997684578857334, −8.409578853511103745454259503108, −8.318272012970896134030915355471, −7.80148911629081451534797706316, −7.58644271444495996613638756312, −7.26252456607141353211515782971, −6.86355015065206450571995951020, −6.65996652849161641384712338128, −6.61067391875904416570933614709, −5.82175593387592209920033989527, −5.79551129756285340068851740203, −5.66318824732485087487866037554, −5.04109463830870922650412143532, −4.88109582633404357410849875738, −4.87616480452632839857544481654, −3.93947098832744351545324993545, −3.93011457200404357013828179400, −3.90153366285414162564750922574, −3.10743837981281914408068596125, −2.86908830676011797181091771367, −1.91027492340035911089610934674, −1.67763356661225371788350724459, −1.01429119232244115145294490384, −0.72488056187630882142132887690,
0.72488056187630882142132887690, 1.01429119232244115145294490384, 1.67763356661225371788350724459, 1.91027492340035911089610934674, 2.86908830676011797181091771367, 3.10743837981281914408068596125, 3.90153366285414162564750922574, 3.93011457200404357013828179400, 3.93947098832744351545324993545, 4.87616480452632839857544481654, 4.88109582633404357410849875738, 5.04109463830870922650412143532, 5.66318824732485087487866037554, 5.79551129756285340068851740203, 5.82175593387592209920033989527, 6.61067391875904416570933614709, 6.65996652849161641384712338128, 6.86355015065206450571995951020, 7.26252456607141353211515782971, 7.58644271444495996613638756312, 7.80148911629081451534797706316, 8.318272012970896134030915355471, 8.409578853511103745454259503108, 8.719332441510926997684578857334, 9.509729547902967316931157586557