L(s) = 1 | + 3.24·2-s − 5.49·3-s + 2.53·4-s − 16.0·5-s − 17.8·6-s + 7·7-s − 17.7·8-s + 3.16·9-s − 52.2·10-s − 13.9·12-s − 35.3·13-s + 22.7·14-s + 88.4·15-s − 77.8·16-s − 40.4·17-s + 10.2·18-s − 118.·19-s − 40.7·20-s − 38.4·21-s − 174.·23-s + 97.4·24-s + 134.·25-s − 114.·26-s + 130.·27-s + 17.7·28-s + 262.·29-s + 286.·30-s + ⋯ |
L(s) = 1 | + 1.14·2-s − 1.05·3-s + 0.316·4-s − 1.43·5-s − 1.21·6-s + 0.377·7-s − 0.784·8-s + 0.117·9-s − 1.65·10-s − 0.334·12-s − 0.754·13-s + 0.433·14-s + 1.52·15-s − 1.21·16-s − 0.577·17-s + 0.134·18-s − 1.42·19-s − 0.455·20-s − 0.399·21-s − 1.58·23-s + 0.828·24-s + 1.07·25-s − 0.865·26-s + 0.933·27-s + 0.119·28-s + 1.68·29-s + 1.74·30-s + ⋯ |
Λ(s)=(=(847s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(847s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
0.5166195726 |
L(21) |
≈ |
0.5166195726 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1−7T |
| 11 | 1 |
good | 2 | 1−3.24T+8T2 |
| 3 | 1+5.49T+27T2 |
| 5 | 1+16.0T+125T2 |
| 13 | 1+35.3T+2.19e3T2 |
| 17 | 1+40.4T+4.91e3T2 |
| 19 | 1+118.T+6.85e3T2 |
| 23 | 1+174.T+1.21e4T2 |
| 29 | 1−262.T+2.43e4T2 |
| 31 | 1+36.1T+2.97e4T2 |
| 37 | 1−19.0T+5.06e4T2 |
| 41 | 1+156.T+6.89e4T2 |
| 43 | 1+287.T+7.95e4T2 |
| 47 | 1−397.T+1.03e5T2 |
| 53 | 1−272.T+1.48e5T2 |
| 59 | 1+507.T+2.05e5T2 |
| 61 | 1+35.5T+2.26e5T2 |
| 67 | 1−979.T+3.00e5T2 |
| 71 | 1−750.T+3.57e5T2 |
| 73 | 1+395.T+3.89e5T2 |
| 79 | 1−736.T+4.93e5T2 |
| 83 | 1+582.T+5.71e5T2 |
| 89 | 1+806.T+7.04e5T2 |
| 97 | 1+957.T+9.12e5T2 |
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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
−10.15364617340704496479691036112, −8.692165156886550043103518769685, −8.086766936372657198370434050822, −6.85499734329122203101798381548, −6.19087026861756053298857249678, −5.11961573762048942432391398474, −4.47912252751872200064768012615, −3.83635299478462737059996578924, −2.48715025033123340933215350948, −0.33210805545570359202129519817,
0.33210805545570359202129519817, 2.48715025033123340933215350948, 3.83635299478462737059996578924, 4.47912252751872200064768012615, 5.11961573762048942432391398474, 6.19087026861756053298857249678, 6.85499734329122203101798381548, 8.086766936372657198370434050822, 8.692165156886550043103518769685, 10.15364617340704496479691036112