L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s + 4·13-s + 3·15-s − 4·17-s + 8·23-s + 4·25-s − 27-s − 7·29-s − 11·31-s + 33-s + 4·37-s − 4·39-s + 4·41-s − 2·43-s − 3·45-s + 2·47-s + 4·51-s − 11·53-s + 3·55-s − 7·59-s − 10·61-s − 12·65-s + 10·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.774·15-s − 0.970·17-s + 1.66·23-s + 4/5·25-s − 0.192·27-s − 1.29·29-s − 1.97·31-s + 0.174·33-s + 0.657·37-s − 0.640·39-s + 0.624·41-s − 0.304·43-s − 0.447·45-s + 0.291·47-s + 0.560·51-s − 1.51·53-s + 0.404·55-s − 0.911·59-s − 1.28·61-s − 1.48·65-s + 1.22·67-s − 0.963·69-s + ⋯ |
Λ(s)=(=(4704s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4704s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.8177441996 |
L(21) |
≈ |
0.8177441996 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+T |
| 7 | 1 |
good | 5 | 1+3T+pT2 |
| 11 | 1+T+pT2 |
| 13 | 1−4T+pT2 |
| 17 | 1+4T+pT2 |
| 19 | 1+pT2 |
| 23 | 1−8T+pT2 |
| 29 | 1+7T+pT2 |
| 31 | 1+11T+pT2 |
| 37 | 1−4T+pT2 |
| 41 | 1−4T+pT2 |
| 43 | 1+2T+pT2 |
| 47 | 1−2T+pT2 |
| 53 | 1+11T+pT2 |
| 59 | 1+7T+pT2 |
| 61 | 1+10T+pT2 |
| 67 | 1−10T+pT2 |
| 71 | 1−6T+pT2 |
| 73 | 1−6T+pT2 |
| 79 | 1−11T+pT2 |
| 83 | 1+11T+pT2 |
| 89 | 1+6T+pT2 |
| 97 | 1+7T+pT2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.196884566769907012523530039680, −7.51155936293345874727212587536, −6.97146397688082967666818404984, −6.14168801835031951132380208285, −5.32244844888582985746142827540, −4.52864700037723516870251514543, −3.81859118303297002295385754293, −3.14314898901266199472959204646, −1.75936318150012083215616202954, −0.51542501758314193126777624873,
0.51542501758314193126777624873, 1.75936318150012083215616202954, 3.14314898901266199472959204646, 3.81859118303297002295385754293, 4.52864700037723516870251514543, 5.32244844888582985746142827540, 6.14168801835031951132380208285, 6.97146397688082967666818404984, 7.51155936293345874727212587536, 8.196884566769907012523530039680