L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s − 11-s + 12-s + 6·13-s + 3·15-s + 16-s − 5·17-s − 18-s + 6·19-s + 3·20-s + 22-s + 5·23-s − 24-s + 4·25-s − 6·26-s + 27-s − 6·29-s − 3·30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.774·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.213·22-s + 1.04·23-s − 0.204·24-s + 4/5·25-s − 1.17·26-s + 0.192·27-s − 1.11·29-s − 0.547·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
Λ(s)=(=(3234s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(3234s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.419660610 |
L(21) |
≈ |
2.419660610 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+T |
| 3 | 1−T |
| 7 | 1 |
| 11 | 1+T |
good | 5 | 1−3T+pT2 |
| 13 | 1−6T+pT2 |
| 17 | 1+5T+pT2 |
| 19 | 1−6T+pT2 |
| 23 | 1−5T+pT2 |
| 29 | 1+6T+pT2 |
| 31 | 1−4T+pT2 |
| 37 | 1+2T+pT2 |
| 41 | 1−5T+pT2 |
| 43 | 1+10T+pT2 |
| 47 | 1−9T+pT2 |
| 53 | 1−2T+pT2 |
| 59 | 1+12T+pT2 |
| 61 | 1+5T+pT2 |
| 67 | 1−5T+pT2 |
| 71 | 1−4T+pT2 |
| 73 | 1−12T+pT2 |
| 79 | 1+T+pT2 |
| 83 | 1−T+pT2 |
| 89 | 1−6T+pT2 |
| 97 | 1−9T+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
−8.842237582261943412497659784450, −8.102581647527097612509052589247, −7.20820344916316378150903617194, −6.46756871669938820534669985051, −5.81123907221281022677407571078, −4.97628229745406794569182790979, −3.69490446332830124495370704994, −2.82218766450276993295029177486, −1.91855494693826267909351961029, −1.09688652548402126508340746432,
1.09688652548402126508340746432, 1.91855494693826267909351961029, 2.82218766450276993295029177486, 3.69490446332830124495370704994, 4.97628229745406794569182790979, 5.81123907221281022677407571078, 6.46756871669938820534669985051, 7.20820344916316378150903617194, 8.102581647527097612509052589247, 8.842237582261943412497659784450