L(s) = 1 | − 2-s + 3·3-s + 4-s − 5-s − 3·6-s + 2·7-s − 8-s + 6·9-s + 10-s − 4·11-s + 3·12-s − 3·13-s − 2·14-s − 3·15-s + 16-s + 17-s − 6·18-s + 3·19-s − 20-s + 6·21-s + 4·22-s − 6·23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 0.688·19-s − 0.223·20-s + 1.30·21-s + 0.852·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s + 0.377·28-s + ⋯ |
Λ(s)=(=(170s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(170s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.336025674 |
L(21) |
≈ |
1.336025674 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+T |
| 5 | 1+T |
| 17 | 1−T |
good | 3 | 1−pT+pT2 |
| 7 | 1−2T+pT2 |
| 11 | 1+4T+pT2 |
| 13 | 1+3T+pT2 |
| 19 | 1−3T+pT2 |
| 23 | 1+6T+pT2 |
| 29 | 1−9T+pT2 |
| 31 | 1+3T+pT2 |
| 37 | 1+8T+pT2 |
| 41 | 1+6T+pT2 |
| 43 | 1−6T+pT2 |
| 47 | 1+13T+pT2 |
| 53 | 1+9T+pT2 |
| 59 | 1−15T+pT2 |
| 61 | 1−7T+pT2 |
| 67 | 1+2T+pT2 |
| 71 | 1−9T+pT2 |
| 73 | 1+3T+pT2 |
| 79 | 1+pT2 |
| 83 | 1−12T+pT2 |
| 89 | 1+9T+pT2 |
| 97 | 1−7T+pT2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.83373639226606959528283639348, −11.77392021354054025159681104411, −10.36366338935225999926837372006, −9.673175371223344610087496327723, −8.338175486309026415938497793980, −8.047148785250349636560357968225, −7.12227398289357983628483830063, −4.92015064777793909582073169694, −3.29460314858197068359451412690, −2.08881038737404835754425957871,
2.08881038737404835754425957871, 3.29460314858197068359451412690, 4.92015064777793909582073169694, 7.12227398289357983628483830063, 8.047148785250349636560357968225, 8.338175486309026415938497793980, 9.673175371223344610087496327723, 10.36366338935225999926837372006, 11.77392021354054025159681104411, 12.83373639226606959528283639348