L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 6·13-s − 16-s − 6·17-s + 19-s + 2·20-s − 4·23-s − 25-s + 6·26-s − 2·29-s − 8·31-s − 5·32-s + 6·34-s − 10·37-s − 38-s − 6·40-s − 2·41-s − 4·43-s + 4·46-s + 12·47-s + 50-s + 6·52-s + 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.229·19-s + 0.447·20-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 1.64·37-s − 0.162·38-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.589·46-s + 1.75·47-s + 0.141·50-s + 0.832·52-s + 0.824·53-s + ⋯ |
Λ(s)=(=(8379s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(8379s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 19 | 1−T |
good | 2 | 1+T+pT2 |
| 5 | 1+2T+pT2 |
| 11 | 1+pT2 |
| 13 | 1+6T+pT2 |
| 17 | 1+6T+pT2 |
| 23 | 1+4T+pT2 |
| 29 | 1+2T+pT2 |
| 31 | 1+8T+pT2 |
| 37 | 1+10T+pT2 |
| 41 | 1+2T+pT2 |
| 43 | 1+4T+pT2 |
| 47 | 1−12T+pT2 |
| 53 | 1−6T+pT2 |
| 59 | 1+12T+pT2 |
| 61 | 1−2T+pT2 |
| 67 | 1+4T+pT2 |
| 71 | 1+pT2 |
| 73 | 1+10T+pT2 |
| 79 | 1+pT2 |
| 83 | 1−16T+pT2 |
| 89 | 1+2T+pT2 |
| 97 | 1+10T+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
−7.34752150402139467995584348966, −6.78418502492955034999702547464, −5.56436473344957075485364042125, −4.94046317313085415521526307990, −4.20789833671809510725213788268, −3.70760760284754853820305649861, −2.47817878081925227300773446057, −1.65980873553023464631625543363, 0, 0,
1.65980873553023464631625543363, 2.47817878081925227300773446057, 3.70760760284754853820305649861, 4.20789833671809510725213788268, 4.94046317313085415521526307990, 5.56436473344957075485364042125, 6.78418502492955034999702547464, 7.34752150402139467995584348966