L(s) = 1 | + 10.2·2-s − 9·3-s + 72.5·4-s − 23.7·5-s − 92.0·6-s + 414.·8-s + 81·9-s − 242.·10-s + 465.·11-s − 652.·12-s + 1.01e3·13-s + 213.·15-s + 1.91e3·16-s − 561.·17-s + 828.·18-s + 1.38e3·19-s − 1.72e3·20-s + 4.75e3·22-s + 4.11e3·23-s − 3.72e3·24-s − 2.56e3·25-s + 1.04e4·26-s − 729·27-s − 2.38e3·29-s + 2.18e3·30-s − 2.95e3·31-s + 6.32e3·32-s + ⋯ |
L(s) = 1 | + 1.80·2-s − 0.577·3-s + 2.26·4-s − 0.424·5-s − 1.04·6-s + 2.28·8-s + 0.333·9-s − 0.767·10-s + 1.16·11-s − 1.30·12-s + 1.67·13-s + 0.245·15-s + 1.87·16-s − 0.471·17-s + 0.602·18-s + 0.881·19-s − 0.963·20-s + 2.09·22-s + 1.62·23-s − 1.32·24-s − 0.819·25-s + 3.02·26-s − 0.192·27-s − 0.525·29-s + 0.443·30-s − 0.551·31-s + 1.09·32-s + ⋯ |
Λ(s)=(=(147s/2ΓC(s)L(s)Λ(6−s)
Λ(s)=(=(147s/2ΓC(s+5/2)L(s)Λ(1−s)
Particular Values
L(3) |
≈ |
5.116554143 |
L(21) |
≈ |
5.116554143 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+9T |
| 7 | 1 |
good | 2 | 1−10.2T+32T2 |
| 5 | 1+23.7T+3.12e3T2 |
| 11 | 1−465.T+1.61e5T2 |
| 13 | 1−1.01e3T+3.71e5T2 |
| 17 | 1+561.T+1.41e6T2 |
| 19 | 1−1.38e3T+2.47e6T2 |
| 23 | 1−4.11e3T+6.43e6T2 |
| 29 | 1+2.38e3T+2.05e7T2 |
| 31 | 1+2.95e3T+2.86e7T2 |
| 37 | 1+9.90e3T+6.93e7T2 |
| 41 | 1−4.47e3T+1.15e8T2 |
| 43 | 1−5.18e3T+1.47e8T2 |
| 47 | 1−3.12e3T+2.29e8T2 |
| 53 | 1−1.14e3T+4.18e8T2 |
| 59 | 1+2.74e4T+7.14e8T2 |
| 61 | 1−2.11e4T+8.44e8T2 |
| 67 | 1+5.55e4T+1.35e9T2 |
| 71 | 1+6.07e3T+1.80e9T2 |
| 73 | 1−1.67e4T+2.07e9T2 |
| 79 | 1+4.84e3T+3.07e9T2 |
| 83 | 1+6.01e4T+3.93e9T2 |
| 89 | 1−6.24e4T+5.58e9T2 |
| 97 | 1−6.36e4T+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
−12.20018803665800357675044940593, −11.39804527996161737958865185391, −10.88704822775999379908833384423, −9.028442259121079944820981210210, −7.28453358243644259375965462763, −6.37342524397915815722611525578, −5.45119488434621746384490922936, −4.18548019338380411579486646028, −3.38392877375504342739088743903, −1.40197763487196187633171187720,
1.40197763487196187633171187720, 3.38392877375504342739088743903, 4.18548019338380411579486646028, 5.45119488434621746384490922936, 6.37342524397915815722611525578, 7.28453358243644259375965462763, 9.028442259121079944820981210210, 10.88704822775999379908833384423, 11.39804527996161737958865185391, 12.20018803665800357675044940593