| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.116554143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.116554143\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 5 | \( 1 + 23.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 465.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.01e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 561.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.38e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.84e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.36e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
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
