| L(s) = 1 | − 0.363·2-s − 2.50·3-s − 1.86·4-s − 1.53·5-s + 0.913·6-s − 1.87·7-s + 1.40·8-s + 3.29·9-s + 0.557·10-s + 11-s + 4.68·12-s + 0.782·13-s + 0.680·14-s + 3.84·15-s + 3.22·16-s − 1.92·17-s − 1.20·18-s − 0.355·19-s + 2.86·20-s + 4.69·21-s − 0.363·22-s + 4.07·23-s − 3.53·24-s − 2.65·25-s − 0.284·26-s − 0.748·27-s + 3.49·28-s + ⋯ |
| L(s) = 1 | − 0.257·2-s − 1.44·3-s − 0.933·4-s − 0.685·5-s + 0.372·6-s − 0.707·7-s + 0.497·8-s + 1.09·9-s + 0.176·10-s + 0.301·11-s + 1.35·12-s + 0.217·13-s + 0.181·14-s + 0.993·15-s + 0.805·16-s − 0.466·17-s − 0.282·18-s − 0.0815·19-s + 0.640·20-s + 1.02·21-s − 0.0775·22-s + 0.849·23-s − 0.720·24-s − 0.530·25-s − 0.0558·26-s − 0.144·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 0.363T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 13 | \( 1 - 0.782T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 + 0.355T + 19T^{2} \) |
| 23 | \( 1 - 4.07T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 - 9.50T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 8.35T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 0.658T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 8.38T + 97T^{2} \) |
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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
−9.186353678789301985863091531637, −8.397395945863854697950993184144, −7.46900096758442805795872559606, −6.49532797552277502847250388143, −5.92280028325903594481791359642, −4.77946917526818582050200851910, −4.33572313892995473405636430500, −3.15816258501196990370114781207, −1.03896257906143815812630632021, 0,
1.03896257906143815812630632021, 3.15816258501196990370114781207, 4.33572313892995473405636430500, 4.77946917526818582050200851910, 5.92280028325903594481791359642, 6.49532797552277502847250388143, 7.46900096758442805795872559606, 8.397395945863854697950993184144, 9.186353678789301985863091531637
