| L(s) = 1 | + 0.517·2-s − 1.41·3-s − 1.73·4-s + 5-s − 0.732·6-s + 0.732·7-s − 1.93·8-s − 0.999·9-s + 0.517·10-s − 5.27·11-s + 2.44·12-s + 1.46·13-s + 0.378·14-s − 1.41·15-s + 2.46·16-s + 6.31·17-s − 0.517·18-s + 4.24·19-s − 1.73·20-s − 1.03·21-s − 2.73·22-s − 8.19·23-s + 2.73·24-s + 25-s + 0.757·26-s + 5.65·27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | + 0.366·2-s − 0.816·3-s − 0.866·4-s + 0.447·5-s − 0.298·6-s + 0.276·7-s − 0.683·8-s − 0.333·9-s + 0.163·10-s − 1.59·11-s + 0.707·12-s + 0.406·13-s + 0.101·14-s − 0.365·15-s + 0.616·16-s + 1.53·17-s − 0.122·18-s + 0.973·19-s − 0.387·20-s − 0.225·21-s − 0.582·22-s − 1.70·23-s + 0.557·24-s + 0.200·25-s + 0.148·26-s + 1.08·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 8.76T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 1.13T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 2.07T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−8.079138062372543684168792319056, −7.42710605202480751322336586539, −6.03437566868495528297008573110, −5.75943892175074466638455777451, −5.20250379698199247263715194308, −4.51308711054977261724012577154, −3.40661485173807791591545838114, −2.65548337108864378230450963148, −1.15983514543131789642637339590, 0,
1.15983514543131789642637339590, 2.65548337108864378230450963148, 3.40661485173807791591545838114, 4.51308711054977261724012577154, 5.20250379698199247263715194308, 5.75943892175074466638455777451, 6.03437566868495528297008573110, 7.42710605202480751322336586539, 8.079138062372543684168792319056
