| L(s) = 1 | − 2.13·2-s + 0.778·3-s + 2.54·4-s − 1.92·5-s − 1.65·6-s − 4.90·7-s − 1.15·8-s − 2.39·9-s + 4.10·10-s − 1.77·11-s + 1.97·12-s + 1.35·13-s + 10.4·14-s − 1.49·15-s − 2.62·16-s + 7.41·17-s + 5.10·18-s + 5.31·19-s − 4.89·20-s − 3.81·21-s + 3.77·22-s + 4.88·23-s − 0.897·24-s − 1.29·25-s − 2.88·26-s − 4.19·27-s − 12.4·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.449·3-s + 1.27·4-s − 0.860·5-s − 0.676·6-s − 1.85·7-s − 0.407·8-s − 0.798·9-s + 1.29·10-s − 0.534·11-s + 0.570·12-s + 0.375·13-s + 2.79·14-s − 0.386·15-s − 0.656·16-s + 1.79·17-s + 1.20·18-s + 1.21·19-s − 1.09·20-s − 0.832·21-s + 0.805·22-s + 1.01·23-s − 0.183·24-s − 0.258·25-s − 0.566·26-s − 0.807·27-s − 2.35·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4428922954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4428922954\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.13T + 2T^{2} \) |
| 3 | \( 1 - 0.778T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 - 8.53T + 29T^{2} \) |
| 31 | \( 1 - 0.671T + 31T^{2} \) |
| 37 | \( 1 + 0.897T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 6.75T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 3.73T + 97T^{2} \) |
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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
−10.26568502845285195958707750823, −9.932874627714027713086294297438, −8.982692914377118254511567622165, −8.284264580664003668994293987517, −7.50714897862041612442702304554, −6.69592356651745519327972718472, −5.45098108870285294062590236605, −3.48190589290341076425595485697, −2.91172084858638317992477876407, −0.70432634347277733047521773286,
0.70432634347277733047521773286, 2.91172084858638317992477876407, 3.48190589290341076425595485697, 5.45098108870285294062590236605, 6.69592356651745519327972718472, 7.50714897862041612442702304554, 8.284264580664003668994293987517, 8.982692914377118254511567622165, 9.932874627714027713086294297438, 10.26568502845285195958707750823
