L(s) = 1 | − 10.3·2-s − 3.25·3-s + 75.7·4-s + 33.7·6-s − 50.5·7-s − 454.·8-s − 232.·9-s − 121·11-s − 246.·12-s − 990.·13-s + 525.·14-s + 2.29e3·16-s − 427.·17-s + 2.41e3·18-s + 2.33e3·19-s + 164.·21-s + 1.25e3·22-s − 4.20e3·23-s + 1.47e3·24-s + 1.02e4·26-s + 1.54e3·27-s − 3.83e3·28-s + 1.54e3·29-s − 6.33e3·31-s − 9.26e3·32-s + 393.·33-s + 4.43e3·34-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 0.208·3-s + 2.36·4-s + 0.383·6-s − 0.390·7-s − 2.51·8-s − 0.956·9-s − 0.301·11-s − 0.494·12-s − 1.62·13-s + 0.716·14-s + 2.24·16-s − 0.358·17-s + 1.75·18-s + 1.48·19-s + 0.0814·21-s + 0.553·22-s − 1.65·23-s + 0.524·24-s + 2.98·26-s + 0.408·27-s − 0.924·28-s + 0.340·29-s − 1.18·31-s − 1.60·32-s + 0.0629·33-s + 0.658·34-s + ⋯ |
Λ(s)=(=(275s/2ΓC(s)L(s)Λ(6−s)
Λ(s)=(=(275s/2ΓC(s+5/2)L(s)Λ(1−s)
Particular Values
L(3) |
≈ |
0.2140989919 |
L(21) |
≈ |
0.2140989919 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 11 | 1+121T |
good | 2 | 1+10.3T+32T2 |
| 3 | 1+3.25T+243T2 |
| 7 | 1+50.5T+1.68e4T2 |
| 13 | 1+990.T+3.71e5T2 |
| 17 | 1+427.T+1.41e6T2 |
| 19 | 1−2.33e3T+2.47e6T2 |
| 23 | 1+4.20e3T+6.43e6T2 |
| 29 | 1−1.54e3T+2.05e7T2 |
| 31 | 1+6.33e3T+2.86e7T2 |
| 37 | 1−1.21e3T+6.93e7T2 |
| 41 | 1+1.49e4T+1.15e8T2 |
| 43 | 1−1.79e3T+1.47e8T2 |
| 47 | 1−1.66e3T+2.29e8T2 |
| 53 | 1+1.19e4T+4.18e8T2 |
| 59 | 1+1.63e4T+7.14e8T2 |
| 61 | 1+6.93e3T+8.44e8T2 |
| 67 | 1+4.86e3T+1.35e9T2 |
| 71 | 1+1.88e4T+1.80e9T2 |
| 73 | 1−6.61e4T+2.07e9T2 |
| 79 | 1−8.30e4T+3.07e9T2 |
| 83 | 1+5.98e3T+3.93e9T2 |
| 89 | 1−4.84e4T+5.58e9T2 |
| 97 | 1+1.09e5T+8.58e9T2 |
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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
−10.76640571532332951063037904829, −9.837214606078333535755446728915, −9.330687308231510067369370362003, −8.162992714569501733990685766102, −7.47485158610776787015991343584, −6.44884762413589452063930017343, −5.27945466485864042904645586997, −3.06434062027717690144530819571, −1.99392290195506979509547184689, −0.33062775347875548383636817176,
0.33062775347875548383636817176, 1.99392290195506979509547184689, 3.06434062027717690144530819571, 5.27945466485864042904645586997, 6.44884762413589452063930017343, 7.47485158610776787015991343584, 8.162992714569501733990685766102, 9.330687308231510067369370362003, 9.837214606078333535755446728915, 10.76640571532332951063037904829