| L(s) = 1 | − 2.30·2-s + 1.47·3-s + 3.30·4-s + 0.847·5-s − 3.39·6-s − 3.00·8-s − 0.829·9-s − 1.95·10-s + 1.50·11-s + 4.86·12-s + 1.24·15-s + 0.313·16-s − 2.07·17-s + 1.90·18-s − 0.0474·19-s + 2.80·20-s − 3.46·22-s − 7.81·23-s − 4.42·24-s − 4.28·25-s − 5.64·27-s + 1.35·29-s − 2.87·30-s + 7.86·31-s + 5.29·32-s + 2.21·33-s + 4.77·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.850·3-s + 1.65·4-s + 0.378·5-s − 1.38·6-s − 1.06·8-s − 0.276·9-s − 0.617·10-s + 0.453·11-s + 1.40·12-s + 0.322·15-s + 0.0782·16-s − 0.502·17-s + 0.450·18-s − 0.0108·19-s + 0.626·20-s − 0.738·22-s − 1.63·23-s − 0.904·24-s − 0.856·25-s − 1.08·27-s + 0.252·29-s − 0.524·30-s + 1.41·31-s + 0.935·32-s + 0.385·33-s + 0.818·34-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 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 - 0.847T + 5T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.0474T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 - 0.360T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 - 1.64T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 0.426T + 97T^{2} \) |
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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
−7.937595304937325907272107618089, −7.06237330712331260210740846349, −6.27007501660022181245427230389, −5.76765668906484446100994747988, −4.46063029163632668358147241590, −3.71316835699486943778863335718, −2.54284264316733230462156043061, −2.19651938016907944512664210813, −1.21099672733928327770836668413, 0,
1.21099672733928327770836668413, 2.19651938016907944512664210813, 2.54284264316733230462156043061, 3.71316835699486943778863335718, 4.46063029163632668358147241590, 5.76765668906484446100994747988, 6.27007501660022181245427230389, 7.06237330712331260210740846349, 7.937595304937325907272107618089
