L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 8·13-s + 5·16-s + 6·17-s − 6·18-s − 4·22-s − 8·24-s − 3·25-s − 16·26-s + 4·27-s + 4·29-s + 8·31-s − 6·32-s + 4·33-s − 12·34-s + 9·36-s − 8·37-s + 16·39-s − 18·41-s + 8·43-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 2.21·13-s + 5/4·16-s + 1.45·17-s − 1.41·18-s − 0.852·22-s − 1.63·24-s − 3/5·25-s − 3.13·26-s + 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.06·32-s + 0.696·33-s − 2.05·34-s + 3/2·36-s − 1.31·37-s + 2.56·39-s − 2.81·41-s + 1.21·43-s + ⋯ |
Λ(s)=(=(10458756s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(10458756s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
10458756
= 22⋅32⋅74⋅112
|
Sign: |
1
|
Analytic conductor: |
666.859 |
Root analytic conductor: |
5.08169 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 10458756, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
3.217111046 |
L(21) |
≈ |
3.217111046 |
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)2 |
| 3 | C1 | (1−T)2 |
| 7 | | 1 |
| 11 | C1 | (1−T)2 |
good | 5 | C22 | 1+3T2+p2T4 |
| 13 | C2 | (1−4T+pT2)2 |
| 17 | C2 | (1−3T+pT2)2 |
| 19 | C22 | 1+10T2+p2T4 |
| 23 | C22 | 1+39T2+p2T4 |
| 29 | C2 | (1−2T+pT2)2 |
| 31 | C2 | (1−4T+pT2)2 |
| 37 | D4 | 1+8T+62T2+8pT3+p2T4 |
| 41 | C2 | (1+9T+pT2)2 |
| 43 | D4 | 1−8T+74T2−8pT3+p2T4 |
| 47 | D4 | 1+8T+pT2+8pT3+p2T4 |
| 53 | C2 | (1−4T+pT2)2 |
| 59 | D4 | 1−8T+22T2−8pT3+p2T4 |
| 61 | D4 | 1+8T+75T2+8pT3+p2T4 |
| 67 | D4 | 1+6T+31T2+6pT3+p2T4 |
| 71 | D4 | 1+16T+178T2+16pT3+p2T4 |
| 73 | D4 | 1−20T+218T2−20pT3+p2T4 |
| 79 | D4 | 1−16T+215T2−16pT3+p2T4 |
| 83 | D4 | 1+10T+79T2+10pT3+p2T4 |
| 89 | D4 | 1+16T+214T2+16pT3+p2T4 |
| 97 | D4 | 1−2T+83T2−2pT3+p2T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.696189294109841831527476580199, −8.470235918383574025210388645234, −8.169971854920359980332266304665, −8.119481213758977546403198644653, −7.43561891862866588715401998641, −7.16554146366174752870743327077, −6.59172544213212610393390989694, −6.55660002192121550067105612549, −5.88194655561563126909706037517, −5.73323795139435005933552708828, −5.01660986466511968432386910326, −4.50655410260769816386160914600, −3.83178771513609563236988613981, −3.60081397328304825178492172226, −3.10975125577073049147902862156, −2.97226777859884400697404589723, −1.92624059830205828812394529498, −1.79510108368002855792555073389, −1.15446190102172941289420198828, −0.74548602570690916536978104487,
0.74548602570690916536978104487, 1.15446190102172941289420198828, 1.79510108368002855792555073389, 1.92624059830205828812394529498, 2.97226777859884400697404589723, 3.10975125577073049147902862156, 3.60081397328304825178492172226, 3.83178771513609563236988613981, 4.50655410260769816386160914600, 5.01660986466511968432386910326, 5.73323795139435005933552708828, 5.88194655561563126909706037517, 6.55660002192121550067105612549, 6.59172544213212610393390989694, 7.16554146366174752870743327077, 7.43561891862866588715401998641, 8.119481213758977546403198644653, 8.169971854920359980332266304665, 8.470235918383574025210388645234, 8.696189294109841831527476580199