L(s) = 1 | − 2·5-s + 4·11-s − 4·13-s + 2·17-s + 4·19-s − 8·23-s + 5·25-s − 12·29-s − 8·31-s − 6·37-s + 12·41-s + 8·43-s − 2·53-s − 8·55-s + 4·59-s + 2·61-s + 8·65-s + 4·67-s − 16·71-s − 10·73-s + 8·79-s + 8·83-s − 4·85-s − 6·89-s − 8·95-s + 4·97-s − 18·101-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 25-s − 2.22·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s + 1.21·43-s − 0.274·53-s − 1.07·55-s + 0.520·59-s + 0.256·61-s + 0.992·65-s + 0.488·67-s − 1.89·71-s − 1.17·73-s + 0.900·79-s + 0.878·83-s − 0.433·85-s − 0.635·89-s − 0.820·95-s + 0.406·97-s − 1.79·101-s + ⋯ |
Λ(s)=(=(12446784s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(12446784s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
12446784
= 26⋅34⋅74
|
Sign: |
1
|
Analytic conductor: |
793.617 |
Root analytic conductor: |
5.30765 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 12446784, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.5412865254 |
L(21) |
≈ |
0.5412865254 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 7 | | 1 |
good | 5 | C22 | 1+2T−T2+2pT3+p2T4 |
| 11 | C22 | 1−4T+5T2−4pT3+p2T4 |
| 13 | C2 | (1+2T+pT2)2 |
| 17 | C22 | 1−2T−13T2−2pT3+p2T4 |
| 19 | C22 | 1−4T−3T2−4pT3+p2T4 |
| 23 | C22 | 1+8T+41T2+8pT3+p2T4 |
| 29 | C2 | (1+6T+pT2)2 |
| 31 | C22 | 1+8T+33T2+8pT3+p2T4 |
| 37 | C22 | 1+6T−T2+6pT3+p2T4 |
| 41 | C2 | (1−6T+pT2)2 |
| 43 | C2 | (1−4T+pT2)2 |
| 47 | C22 | 1−pT2+p2T4 |
| 53 | C22 | 1+2T−49T2+2pT3+p2T4 |
| 59 | C22 | 1−4T−43T2−4pT3+p2T4 |
| 61 | C22 | 1−2T−57T2−2pT3+p2T4 |
| 67 | C22 | 1−4T−51T2−4pT3+p2T4 |
| 71 | C2 | (1+8T+pT2)2 |
| 73 | C2 | (1−7T+pT2)(1+17T+pT2) |
| 79 | C22 | 1−8T−15T2−8pT3+p2T4 |
| 83 | C2 | (1−4T+pT2)2 |
| 89 | C22 | 1+6T−53T2+6pT3+p2T4 |
| 97 | C2 | (1−2T+pT2)2 |
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
−9.204046596487589180186904514203, −8.272553344426047313612471424621, −7.85512773338797021093032314339, −7.59683149763123796779310282940, −7.45974219323980043518733786024, −6.90630717232796421988368871476, −6.76247297475051832343364194177, −5.96531850397358392662028126581, −5.84162205342835299652013332632, −5.24370443692848804349558117651, −5.16727025592672306048199659099, −4.25537204051607515531005631063, −4.14836954244510441850702376959, −3.66051715115320171014303449147, −3.56105981485770482564089602739, −2.57539903730649162076855502659, −2.51177880333869842261581254860, −1.54596178224669240159256437825, −1.31174783387325350265933982568, −0.22571247862950536842039322377,
0.22571247862950536842039322377, 1.31174783387325350265933982568, 1.54596178224669240159256437825, 2.51177880333869842261581254860, 2.57539903730649162076855502659, 3.56105981485770482564089602739, 3.66051715115320171014303449147, 4.14836954244510441850702376959, 4.25537204051607515531005631063, 5.16727025592672306048199659099, 5.24370443692848804349558117651, 5.84162205342835299652013332632, 5.96531850397358392662028126581, 6.76247297475051832343364194177, 6.90630717232796421988368871476, 7.45974219323980043518733786024, 7.59683149763123796779310282940, 7.85512773338797021093032314339, 8.272553344426047313612471424621, 9.204046596487589180186904514203