L(s) = 1 | − 1.35·2-s + 0.880·3-s − 0.162·4-s + 5-s − 1.19·6-s − 3.54·7-s + 2.93·8-s − 2.22·9-s − 1.35·10-s − 6.04·11-s − 0.142·12-s − 0.780·13-s + 4.81·14-s + 0.880·15-s − 3.64·16-s − 3.48·17-s + 3.01·18-s − 4.59·19-s − 0.162·20-s − 3.12·21-s + 8.19·22-s + 2.96·23-s + 2.58·24-s + 25-s + 1.05·26-s − 4.59·27-s + 0.575·28-s + ⋯ |
L(s) = 1 | − 0.958·2-s + 0.508·3-s − 0.0810·4-s + 0.447·5-s − 0.487·6-s − 1.34·7-s + 1.03·8-s − 0.741·9-s − 0.428·10-s − 1.82·11-s − 0.0412·12-s − 0.216·13-s + 1.28·14-s + 0.227·15-s − 0.912·16-s − 0.844·17-s + 0.710·18-s − 1.05·19-s − 0.0362·20-s − 0.681·21-s + 1.74·22-s + 0.619·23-s + 0.526·24-s + 0.200·25-s + 0.207·26-s − 0.885·27-s + 0.108·28-s + ⋯ |
Λ(s)=(=(4805s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4805s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.2507566425 |
L(21) |
≈ |
0.2507566425 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1−T |
| 31 | 1 |
good | 2 | 1+1.35T+2T2 |
| 3 | 1−0.880T+3T2 |
| 7 | 1+3.54T+7T2 |
| 11 | 1+6.04T+11T2 |
| 13 | 1+0.780T+13T2 |
| 17 | 1+3.48T+17T2 |
| 19 | 1+4.59T+19T2 |
| 23 | 1−2.96T+23T2 |
| 29 | 1−5.27T+29T2 |
| 37 | 1+4.63T+37T2 |
| 41 | 1+7.86T+41T2 |
| 43 | 1−0.825T+43T2 |
| 47 | 1+11.6T+47T2 |
| 53 | 1+9.04T+53T2 |
| 59 | 1−12.3T+59T2 |
| 61 | 1+1.81T+61T2 |
| 67 | 1+7.06T+67T2 |
| 71 | 1−15.7T+71T2 |
| 73 | 1−1.06T+73T2 |
| 79 | 1+12.1T+79T2 |
| 83 | 1−11.8T+83T2 |
| 89 | 1+3.40T+89T2 |
| 97 | 1+5.51T+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
−8.472304535079587586798103479336, −7.85975606883891755581623996192, −6.90781242163075747140380259654, −6.35313913080618449420902668362, −5.30392541679114789270573352950, −4.70809712963102967872389005424, −3.44243117071066128575369892462, −2.72070290754976200306725100819, −2.00269884508904230506046073489, −0.28867199182917198821895248717,
0.28867199182917198821895248717, 2.00269884508904230506046073489, 2.72070290754976200306725100819, 3.44243117071066128575369892462, 4.70809712963102967872389005424, 5.30392541679114789270573352950, 6.35313913080618449420902668362, 6.90781242163075747140380259654, 7.85975606883891755581623996192, 8.472304535079587586798103479336