L(s) = 1 | − 2.21·2-s + 2.88·4-s + 1.31·7-s − 1.96·8-s + 4.29·11-s + 2.97·13-s − 2.91·14-s − 1.43·16-s + 0.642·17-s − 5.07·19-s − 9.48·22-s + 8.84·23-s − 6.57·26-s + 3.80·28-s − 29-s − 6.27·31-s + 7.10·32-s − 1.42·34-s − 0.934·37-s + 11.2·38-s + 11.0·41-s − 2.03·43-s + 12.3·44-s − 19.5·46-s + 9.57·47-s − 5.26·49-s + 8.59·52-s + ⋯ |
L(s) = 1 | − 1.56·2-s + 1.44·4-s + 0.497·7-s − 0.693·8-s + 1.29·11-s + 0.825·13-s − 0.777·14-s − 0.359·16-s + 0.155·17-s − 1.16·19-s − 2.02·22-s + 1.84·23-s − 1.29·26-s + 0.718·28-s − 0.185·29-s − 1.12·31-s + 1.25·32-s − 0.243·34-s − 0.153·37-s + 1.82·38-s + 1.73·41-s − 0.311·43-s + 1.86·44-s − 2.88·46-s + 1.39·47-s − 0.752·49-s + 1.19·52-s + ⋯ |
Λ(s)=(=(6525s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(6525s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.149480921 |
L(21) |
≈ |
1.149480921 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
| 29 | 1+T |
good | 2 | 1+2.21T+2T2 |
| 7 | 1−1.31T+7T2 |
| 11 | 1−4.29T+11T2 |
| 13 | 1−2.97T+13T2 |
| 17 | 1−0.642T+17T2 |
| 19 | 1+5.07T+19T2 |
| 23 | 1−8.84T+23T2 |
| 31 | 1+6.27T+31T2 |
| 37 | 1+0.934T+37T2 |
| 41 | 1−11.0T+41T2 |
| 43 | 1+2.03T+43T2 |
| 47 | 1−9.57T+47T2 |
| 53 | 1+13.0T+53T2 |
| 59 | 1+2.55T+59T2 |
| 61 | 1−8.46T+61T2 |
| 67 | 1−12.9T+67T2 |
| 71 | 1−5.10T+71T2 |
| 73 | 1−16.0T+73T2 |
| 79 | 1+3.30T+79T2 |
| 83 | 1−5.24T+83T2 |
| 89 | 1+5.05T+89T2 |
| 97 | 1−2.04T+97T2 |
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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
−8.159426086225580764521116838056, −7.50063298534563337183898801465, −6.73250924967370053620396349675, −6.32361593390043195202639683875, −5.24515344907496627410145832019, −4.30685404334794276469933974745, −3.53334614313887340554896238582, −2.32016921242010620489014593874, −1.48610245672610154005068073947, −0.77582523477395432958171402416,
0.77582523477395432958171402416, 1.48610245672610154005068073947, 2.32016921242010620489014593874, 3.53334614313887340554896238582, 4.30685404334794276469933974745, 5.24515344907496627410145832019, 6.32361593390043195202639683875, 6.73250924967370053620396349675, 7.50063298534563337183898801465, 8.159426086225580764521116838056