| L(s) = 1 | − 2-s − 1.27·3-s + 4-s − 2.37·5-s + 1.27·6-s − 0.273·7-s − 8-s − 1.37·9-s + 2.37·10-s + 4.37·11-s − 1.27·12-s + 0.273·14-s + 3.02·15-s + 16-s + 0.547·17-s + 1.37·18-s − 19-s − 2.37·20-s + 0.348·21-s − 4.37·22-s − 4.57·23-s + 1.27·24-s + 0.651·25-s + 5.57·27-s − 0.273·28-s + 1.67·29-s − 3.02·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.735·3-s + 0.5·4-s − 1.06·5-s + 0.520·6-s − 0.103·7-s − 0.353·8-s − 0.459·9-s + 0.751·10-s + 1.31·11-s − 0.367·12-s + 0.0732·14-s + 0.781·15-s + 0.250·16-s + 0.132·17-s + 0.324·18-s − 0.229·19-s − 0.531·20-s + 0.0761·21-s − 0.933·22-s − 0.954·23-s + 0.260·24-s + 0.130·25-s + 1.07·27-s − 0.0517·28-s + 0.311·29-s − 0.552·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 0.273T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 0.547T + 17T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 0.452T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 2.36T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−7.86855939716069019323982474708, −6.85716497869784616587706242644, −6.45561807319838471800427829250, −5.73130386801674108895980679124, −4.79040806918424205654959949302, −3.92395533145189972565154304717, −3.32035582512412117948066313204, −2.10177373209538311474653444348, −0.943000181828108507209970197342, 0,
0.943000181828108507209970197342, 2.10177373209538311474653444348, 3.32035582512412117948066313204, 3.92395533145189972565154304717, 4.79040806918424205654959949302, 5.73130386801674108895980679124, 6.45561807319838471800427829250, 6.85716497869784616587706242644, 7.86855939716069019323982474708
