| L(s) = 1 | + 3.77·2-s − 3·3-s + 6.25·4-s + 16.4·5-s − 11.3·6-s − 8.52·7-s − 6.60·8-s + 9·9-s + 62.1·10-s − 11·11-s − 18.7·12-s − 45.6·13-s − 32.1·14-s − 49.3·15-s − 74.9·16-s + 130.·17-s + 33.9·18-s − 78.9·19-s + 102.·20-s + 25.5·21-s − 41.5·22-s − 34.6·23-s + 19.8·24-s + 146.·25-s − 172.·26-s − 27·27-s − 53.2·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.781·4-s + 1.47·5-s − 0.770·6-s − 0.460·7-s − 0.291·8-s + 0.333·9-s + 1.96·10-s − 0.301·11-s − 0.451·12-s − 0.973·13-s − 0.614·14-s − 0.850·15-s − 1.17·16-s + 1.86·17-s + 0.444·18-s − 0.952·19-s + 1.15·20-s + 0.265·21-s − 0.402·22-s − 0.314·23-s + 0.168·24-s + 1.16·25-s − 1.29·26-s − 0.192·27-s − 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 3.77T + 8T^{2} \) |
| 5 | \( 1 - 16.4T + 125T^{2} \) |
| 7 | \( 1 + 8.52T + 343T^{2} \) |
| 13 | \( 1 + 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 84.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 323.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 1.70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 216.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 641.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 848.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 122.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 349.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 47.2T + 9.12e5T^{2} \) |
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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
−8.407482044808663413044838800471, −7.20324425754974754307361340957, −6.35995065077076540379104299517, −5.90594238436950225784242744315, −5.18976963167747099710943552979, −4.64706480605066640768916512110, −3.38498884084267591741194997667, −2.61717212831886942328566410471, −1.56959236596183102793918036991, 0,
1.56959236596183102793918036991, 2.61717212831886942328566410471, 3.38498884084267591741194997667, 4.64706480605066640768916512110, 5.18976963167747099710943552979, 5.90594238436950225784242744315, 6.35995065077076540379104299517, 7.20324425754974754307361340957, 8.407482044808663413044838800471
