L(s) = 1 | + 1.21·2-s + 1.31·3-s − 0.525·4-s + 1.59·6-s − 2.90·7-s − 3.06·8-s − 1.28·9-s − 0.214·11-s − 0.688·12-s − 3.52·14-s − 2.67·16-s + 6.42·17-s − 1.55·18-s − 2.21·19-s − 3.80·21-s − 0.260·22-s + 4.68·23-s − 4.02·24-s − 5.61·27-s + 1.52·28-s + 8.70·29-s + 5.59·31-s + 2.88·32-s − 0.280·33-s + 7.80·34-s + 0.673·36-s − 2.28·37-s + ⋯ |
L(s) = 1 | + 0.858·2-s + 0.756·3-s − 0.262·4-s + 0.649·6-s − 1.09·7-s − 1.08·8-s − 0.426·9-s − 0.0646·11-s − 0.198·12-s − 0.942·14-s − 0.668·16-s + 1.55·17-s − 0.366·18-s − 0.507·19-s − 0.830·21-s − 0.0554·22-s + 0.977·23-s − 0.820·24-s − 1.08·27-s + 0.288·28-s + 1.61·29-s + 1.00·31-s + 0.510·32-s − 0.0489·33-s + 1.33·34-s + 0.112·36-s − 0.374·37-s + ⋯ |
Λ(s)=(=(4225s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4225s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.538041088 |
L(21) |
≈ |
2.538041088 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 13 | 1 |
good | 2 | 1−1.21T+2T2 |
| 3 | 1−1.31T+3T2 |
| 7 | 1+2.90T+7T2 |
| 11 | 1+0.214T+11T2 |
| 17 | 1−6.42T+17T2 |
| 19 | 1+2.21T+19T2 |
| 23 | 1−4.68T+23T2 |
| 29 | 1−8.70T+29T2 |
| 31 | 1−5.59T+31T2 |
| 37 | 1+2.28T+37T2 |
| 41 | 1+3.05T+41T2 |
| 43 | 1−6.36T+43T2 |
| 47 | 1+1.09T+47T2 |
| 53 | 1+6.23T+53T2 |
| 59 | 1−9.26T+59T2 |
| 61 | 1+0.280T+61T2 |
| 67 | 1−7.76T+67T2 |
| 71 | 1−6.08T+71T2 |
| 73 | 1−10.2T+73T2 |
| 79 | 1+14.2T+79T2 |
| 83 | 1+9.52T+83T2 |
| 89 | 1−5.61T+89T2 |
| 97 | 1−18.0T+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
−8.515788666886452674183668495883, −7.76700154372687429834130477561, −6.71487164063781241868307298241, −6.12996314920509896819912545903, −5.36784393426314463318916251015, −4.60441166909348913119623895609, −3.56281971992993236525387143163, −3.17615478558072498665926290285, −2.50103964926629481594348090728, −0.74836573588688376833158998024,
0.74836573588688376833158998024, 2.50103964926629481594348090728, 3.17615478558072498665926290285, 3.56281971992993236525387143163, 4.60441166909348913119623895609, 5.36784393426314463318916251015, 6.12996314920509896819912545903, 6.71487164063781241868307298241, 7.76700154372687429834130477561, 8.515788666886452674183668495883