L(s) = 1 | − 5-s + 1.26·7-s + 5.46·11-s + 4·13-s + 4·17-s + 6.19·19-s + 8.92·23-s + 25-s − 7.46·29-s + 10.1·31-s − 1.26·35-s + 37-s − 4.92·41-s + 6·43-s − 1.26·47-s − 5.39·49-s − 12.9·53-s − 5.46·55-s − 3.66·59-s + 14·61-s − 4·65-s − 5.26·67-s − 8·71-s − 10.3·73-s + 6.92·77-s + 6.19·79-s − 1.66·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.479·7-s + 1.64·11-s + 1.10·13-s + 0.970·17-s + 1.42·19-s + 1.86·23-s + 0.200·25-s − 1.38·29-s + 1.83·31-s − 0.214·35-s + 0.164·37-s − 0.769·41-s + 0.914·43-s − 0.184·47-s − 0.770·49-s − 1.77·53-s − 0.736·55-s − 0.476·59-s + 1.79·61-s − 0.496·65-s − 0.643·67-s − 0.949·71-s − 1.21·73-s + 0.789·77-s + 0.697·79-s − 0.182·83-s + ⋯ |
Λ(s)=(=(6660s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(6660s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.850563865 |
L(21) |
≈ |
2.850563865 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+T |
| 37 | 1−T |
good | 7 | 1−1.26T+7T2 |
| 11 | 1−5.46T+11T2 |
| 13 | 1−4T+13T2 |
| 17 | 1−4T+17T2 |
| 19 | 1−6.19T+19T2 |
| 23 | 1−8.92T+23T2 |
| 29 | 1+7.46T+29T2 |
| 31 | 1−10.1T+31T2 |
| 41 | 1+4.92T+41T2 |
| 43 | 1−6T+43T2 |
| 47 | 1+1.26T+47T2 |
| 53 | 1+12.9T+53T2 |
| 59 | 1+3.66T+59T2 |
| 61 | 1−14T+61T2 |
| 67 | 1+5.26T+67T2 |
| 71 | 1+8T+71T2 |
| 73 | 1+10.3T+73T2 |
| 79 | 1−6.19T+79T2 |
| 83 | 1+1.66T+83T2 |
| 89 | 1−2T+89T2 |
| 97 | 1+14T+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
−7.985796035268412574264512571425, −7.30003461503997691114460293543, −6.64194599837109156851624099607, −5.90989875910007022026534618012, −5.11063074685569888081893924172, −4.34665405293149061196932876670, −3.51683620110618350610943882838, −3.04029320729344446829243626781, −1.40885739769473434892341782349, −1.06571257106687043127735882754,
1.06571257106687043127735882754, 1.40885739769473434892341782349, 3.04029320729344446829243626781, 3.51683620110618350610943882838, 4.34665405293149061196932876670, 5.11063074685569888081893924172, 5.90989875910007022026534618012, 6.64194599837109156851624099607, 7.30003461503997691114460293543, 7.985796035268412574264512571425