| L(s) = 1 | − 2-s + 1.17·3-s + 4-s − 2.97·5-s − 1.17·6-s + 0.520·7-s − 8-s − 1.63·9-s + 2.97·10-s + 0.649·11-s + 1.17·12-s + 0.493·13-s − 0.520·14-s − 3.47·15-s + 16-s + 7.42·17-s + 1.63·18-s − 6.63·19-s − 2.97·20-s + 0.609·21-s − 0.649·22-s + 7.61·23-s − 1.17·24-s + 3.83·25-s − 0.493·26-s − 5.41·27-s + 0.520·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.675·3-s + 0.5·4-s − 1.32·5-s − 0.477·6-s + 0.196·7-s − 0.353·8-s − 0.543·9-s + 0.939·10-s + 0.195·11-s + 0.337·12-s + 0.136·13-s − 0.139·14-s − 0.898·15-s + 0.250·16-s + 1.79·17-s + 0.384·18-s − 1.52·19-s − 0.664·20-s + 0.133·21-s − 0.138·22-s + 1.58·23-s − 0.238·24-s + 0.767·25-s − 0.0968·26-s − 1.04·27-s + 0.0984·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 - 0.520T + 7T^{2} \) |
| 11 | \( 1 - 0.649T + 11T^{2} \) |
| 13 | \( 1 - 0.493T + 13T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 0.649T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 + 1.15T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 - 4.52T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.74T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.759007164246351629550128277690, −8.201648913282352008155985608083, −7.68212697297548071118816545924, −6.89736417981730844087748892395, −5.81078215618826172126343847436, −4.67782260490590360790796398911, −3.52333116883364769411282981019, −3.02697235093668118524799738915, −1.55213401277962898809576348988, 0,
1.55213401277962898809576348988, 3.02697235093668118524799738915, 3.52333116883364769411282981019, 4.67782260490590360790796398911, 5.81078215618826172126343847436, 6.89736417981730844087748892395, 7.68212697297548071118816545924, 8.201648913282352008155985608083, 8.759007164246351629550128277690
