| L(s) = 1 | + 1.94·2-s − 1.73·3-s + 1.79·4-s − 3.71·5-s − 3.37·6-s − 0.406·8-s + 0.00865·9-s − 7.24·10-s + 3.56·11-s − 3.10·12-s + 6.45·15-s − 4.37·16-s + 1.69·17-s + 0.0168·18-s − 5.49·19-s − 6.66·20-s + 6.93·22-s + 6.19·23-s + 0.705·24-s + 8.83·25-s + 5.18·27-s − 1.39·29-s + 12.5·30-s + 7.04·31-s − 7.70·32-s − 6.18·33-s + 3.29·34-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 1.00·3-s + 0.895·4-s − 1.66·5-s − 1.37·6-s − 0.143·8-s + 0.00288·9-s − 2.29·10-s + 1.07·11-s − 0.896·12-s + 1.66·15-s − 1.09·16-s + 0.411·17-s + 0.00397·18-s − 1.26·19-s − 1.48·20-s + 1.47·22-s + 1.29·23-s + 0.143·24-s + 1.76·25-s + 0.998·27-s − 0.258·29-s + 2.29·30-s + 1.26·31-s − 1.36·32-s − 1.07·33-s + 0.565·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 - 0.877T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 - 4.04T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 - 1.44T + 83T^{2} \) |
| 89 | \( 1 + 4.46T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.08182030517503190606540522953, −6.61273308032075706230580462630, −5.99805730698798674261279795016, −5.24521594053840398669343572099, −4.41792994773349860044434808768, −4.24536756267906533027208368533, −3.37845979215154090767798224808, −2.69243628899403744819095162697, −1.04812428001961829324790428954, 0,
1.04812428001961829324790428954, 2.69243628899403744819095162697, 3.37845979215154090767798224808, 4.24536756267906533027208368533, 4.41792994773349860044434808768, 5.24521594053840398669343572099, 5.99805730698798674261279795016, 6.61273308032075706230580462630, 7.08182030517503190606540522953
