| L(s) = 1 | − 2.25·2-s − 1.07·3-s + 3.10·4-s − 0.586·5-s + 2.43·6-s + 0.547·7-s − 2.50·8-s − 1.84·9-s + 1.32·10-s + 3.84·11-s − 3.34·12-s − 2.69·13-s − 1.23·14-s + 0.630·15-s − 0.560·16-s + 2.87·17-s + 4.16·18-s − 4.35·19-s − 1.82·20-s − 0.589·21-s − 8.67·22-s + 5.43·23-s + 2.69·24-s − 4.65·25-s + 6.09·26-s + 5.21·27-s + 1.70·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.621·3-s + 1.55·4-s − 0.262·5-s + 0.992·6-s + 0.206·7-s − 0.884·8-s − 0.614·9-s + 0.418·10-s + 1.15·11-s − 0.965·12-s − 0.747·13-s − 0.330·14-s + 0.162·15-s − 0.140·16-s + 0.697·17-s + 0.981·18-s − 0.998·19-s − 0.407·20-s − 0.128·21-s − 1.85·22-s + 1.13·23-s + 0.549·24-s − 0.931·25-s + 1.19·26-s + 1.00·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4520166423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4520166423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 1.07T + 3T^{2} \) |
| 5 | \( 1 + 0.586T + 5T^{2} \) |
| 7 | \( 1 - 0.547T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 - 6.06T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 7.76T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 2.86T + 97T^{2} \) |
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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
−10.89292436325407057452966980608, −9.688990895090546338583023808987, −9.175049299952916431798645519606, −8.222721791484132424169128347695, −7.41355912273124279565987916462, −6.53075840958287746819317234777, −5.49370418499592928597057024882, −4.06155453198951231027091834092, −2.32106285996061779523099000883, −0.77942525476863621532807966126,
0.77942525476863621532807966126, 2.32106285996061779523099000883, 4.06155453198951231027091834092, 5.49370418499592928597057024882, 6.53075840958287746819317234777, 7.41355912273124279565987916462, 8.222721791484132424169128347695, 9.175049299952916431798645519606, 9.688990895090546338583023808987, 10.89292436325407057452966980608
