| L(s) = 1 | − 2-s − 0.956·3-s + 4-s − 0.0953·5-s + 0.956·6-s − 4.20·7-s − 8-s − 2.08·9-s + 0.0953·10-s − 0.121·11-s − 0.956·12-s − 1.63·13-s + 4.20·14-s + 0.0911·15-s + 16-s + 3.22·17-s + 2.08·18-s − 1.33·19-s − 0.0953·20-s + 4.02·21-s + 0.121·22-s + 7.51·23-s + 0.956·24-s − 4.99·25-s + 1.63·26-s + 4.86·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.552·3-s + 0.5·4-s − 0.0426·5-s + 0.390·6-s − 1.58·7-s − 0.353·8-s − 0.694·9-s + 0.0301·10-s − 0.0366·11-s − 0.276·12-s − 0.453·13-s + 1.12·14-s + 0.0235·15-s + 0.250·16-s + 0.781·17-s + 0.491·18-s − 0.307·19-s − 0.0213·20-s + 0.878·21-s + 0.0259·22-s + 1.56·23-s + 0.195·24-s − 0.998·25-s + 0.320·26-s + 0.936·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 0.956T + 3T^{2} \) |
| 5 | \( 1 + 0.0953T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 0.121T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 7.51T + 23T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 + 0.847T + 37T^{2} \) |
| 41 | \( 1 + 2.51T + 41T^{2} \) |
| 43 | \( 1 - 6.69T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 - 0.786T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−8.135530260218022035547418575549, −7.24893646284680643260603239676, −6.66184853418188082744164916439, −5.97624789093461912837587342322, −5.39114047057030885342063642568, −4.22772183149786087778981095997, −3.04463585073104024336303867832, −2.69107197553148783513230364722, −1.00080182422731033960325648561, 0,
1.00080182422731033960325648561, 2.69107197553148783513230364722, 3.04463585073104024336303867832, 4.22772183149786087778981095997, 5.39114047057030885342063642568, 5.97624789093461912837587342322, 6.66184853418188082744164916439, 7.24893646284680643260603239676, 8.135530260218022035547418575549
