L(s) = 1 | − 2·5-s − 12·13-s + 2·17-s + 4·19-s − 4·23-s + 5·25-s + 20·29-s − 8·31-s − 6·37-s − 4·41-s − 8·43-s − 8·47-s − 10·53-s − 12·59-s − 2·61-s + 24·65-s − 12·67-s + 24·71-s − 14·73-s + 8·79-s + 24·83-s − 4·85-s + 2·89-s − 8·95-s − 20·97-s + 6·101-s + 2·109-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 3.32·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 25-s + 3.71·29-s − 1.43·31-s − 0.986·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s − 1.37·53-s − 1.56·59-s − 0.256·61-s + 2.97·65-s − 1.46·67-s + 2.84·71-s − 1.63·73-s + 0.900·79-s + 2.63·83-s − 0.433·85-s + 0.211·89-s − 0.820·95-s − 2.03·97-s + 0.597·101-s + 0.191·109-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) |
≈ |
0.2566383547 |
L(21) |
≈ |
0.2566383547 |
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−pT2+p2T4 |
| 13 | C2 | (1+6T+pT2)2 |
| 17 | C22 | 1−2T−13T2−2pT3+p2T4 |
| 19 | C22 | 1−4T−3T2−4pT3+p2T4 |
| 23 | C22 | 1+4T−7T2+4pT3+p2T4 |
| 29 | C2 | (1−10T+pT2)2 |
| 31 | C22 | 1+8T+33T2+8pT3+p2T4 |
| 37 | C22 | 1+6T−T2+6pT3+p2T4 |
| 41 | C2 | (1+2T+pT2)2 |
| 43 | C2 | (1+4T+pT2)2 |
| 47 | C22 | 1+8T+17T2+8pT3+p2T4 |
| 53 | C22 | 1+10T+47T2+10pT3+p2T4 |
| 59 | C22 | 1+12T+85T2+12pT3+p2T4 |
| 61 | C22 | 1+2T−57T2+2pT3+p2T4 |
| 67 | C22 | 1+12T+77T2+12pT3+p2T4 |
| 71 | C2 | (1−12T+pT2)2 |
| 73 | C22 | 1+14T+123T2+14pT3+p2T4 |
| 79 | C22 | 1−8T−15T2−8pT3+p2T4 |
| 83 | C2 | (1−12T+pT2)2 |
| 89 | C22 | 1−2T−85T2−2pT3+p2T4 |
| 97 | C2 | (1+10T+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
−8.640999272341286951049326553436, −8.154117198051418666831477395359, −8.039157493234952738651435269704, −7.71084010956113871197564615937, −7.31666385090932706879795966349, −6.86697632945751690128319984656, −6.73219651372610855731367190477, −6.33412977954345192539242047627, −5.59848273141288266590724538708, −5.13972949171431704581934301483, −4.90874190317743168797615358570, −4.65586883738171795913397678717, −4.37328711784558422094157111598, −3.44944934689807826946092778836, −3.26544837931114062552187486371, −2.88148956732844158701413511855, −2.35005770868575698380149320852, −1.81349383031692935641239444720, −1.08633607958420632713005377265, −0.16400404974601194804692622465,
0.16400404974601194804692622465, 1.08633607958420632713005377265, 1.81349383031692935641239444720, 2.35005770868575698380149320852, 2.88148956732844158701413511855, 3.26544837931114062552187486371, 3.44944934689807826946092778836, 4.37328711784558422094157111598, 4.65586883738171795913397678717, 4.90874190317743168797615358570, 5.13972949171431704581934301483, 5.59848273141288266590724538708, 6.33412977954345192539242047627, 6.73219651372610855731367190477, 6.86697632945751690128319984656, 7.31666385090932706879795966349, 7.71084010956113871197564615937, 8.039157493234952738651435269704, 8.154117198051418666831477395359, 8.640999272341286951049326553436