L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s − 2·11-s + 12-s + 5·13-s + 14-s − 16-s − 4·17-s + 18-s − 21-s − 2·22-s − 4·23-s + 3·24-s − 5·25-s + 5·26-s − 27-s − 28-s − 8·29-s − 3·31-s + 5·32-s + 2·33-s − 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 25-s + 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.48·29-s − 0.538·31-s + 0.883·32-s + 0.348·33-s − 0.685·34-s + ⋯ |
Λ(s)=(=(1083s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(1083s/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+T |
| 19 | 1 |
good | 2 | 1−T+pT2 |
| 5 | 1+pT2 |
| 7 | 1−T+pT2 |
| 11 | 1+2T+pT2 |
| 13 | 1−5T+pT2 |
| 17 | 1+4T+pT2 |
| 23 | 1+4T+pT2 |
| 29 | 1+8T+pT2 |
| 31 | 1+3T+pT2 |
| 37 | 1−3T+pT2 |
| 41 | 1+12T+pT2 |
| 43 | 1+T+pT2 |
| 47 | 1+6T+pT2 |
| 53 | 1−4T+pT2 |
| 59 | 1−10T+pT2 |
| 61 | 1+13T+pT2 |
| 67 | 1−11T+pT2 |
| 71 | 1−6T+pT2 |
| 73 | 1+11T+pT2 |
| 79 | 1−T+pT2 |
| 83 | 1+pT2 |
| 89 | 1+6T+pT2 |
| 97 | 1−2T+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
−9.490859406317181910865214053483, −8.595490268432358233091447782559, −7.88766191356537491665576331142, −6.62079287213053781803056519115, −5.84047896161452237997455144358, −5.18046014163099724265079503817, −4.20637327455860491087854991343, −3.48638822023419989452758004968, −1.86277933111965985175848882827, 0,
1.86277933111965985175848882827, 3.48638822023419989452758004968, 4.20637327455860491087854991343, 5.18046014163099724265079503817, 5.84047896161452237997455144358, 6.62079287213053781803056519115, 7.88766191356537491665576331142, 8.595490268432358233091447782559, 9.490859406317181910865214053483