| L(s) = 1 | − 1.20·2-s + 3-s − 0.554·4-s − 0.710·5-s − 1.20·6-s − 7-s + 3.07·8-s + 9-s + 0.854·10-s + 4.29·11-s − 0.554·12-s − 0.453·13-s + 1.20·14-s − 0.710·15-s − 2.58·16-s + 2.81·17-s − 1.20·18-s − 4.03·19-s + 0.394·20-s − 21-s − 5.15·22-s + 0.134·23-s + 3.07·24-s − 4.49·25-s + 0.545·26-s + 27-s + 0.554·28-s + ⋯ |
| L(s) = 1 | − 0.850·2-s + 0.577·3-s − 0.277·4-s − 0.317·5-s − 0.490·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s + 0.270·10-s + 1.29·11-s − 0.160·12-s − 0.125·13-s + 0.321·14-s − 0.183·15-s − 0.645·16-s + 0.683·17-s − 0.283·18-s − 0.925·19-s + 0.0881·20-s − 0.218·21-s − 1.09·22-s + 0.0279·23-s + 0.626·24-s − 0.898·25-s + 0.106·26-s + 0.192·27-s + 0.104·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 5 | \( 1 + 0.710T + 5T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 + 0.453T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 + 4.03T + 19T^{2} \) |
| 23 | \( 1 - 0.134T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 8.49T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 + 8.16T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.247T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 - 0.630T + 67T^{2} \) |
| 71 | \( 1 - 0.205T + 71T^{2} \) |
| 73 | \( 1 + 2.56T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.74285862214245067124644104735, −6.89246668485072600270343561885, −6.44169037794429369403741529717, −5.34199039125556934636135680191, −4.44523923504762349000224992800, −3.88045360966584945360484972547, −3.18024791988869359604372995121, −1.96132889018229193136628811417, −1.19837892921116467566047950567, 0,
1.19837892921116467566047950567, 1.96132889018229193136628811417, 3.18024791988869359604372995121, 3.88045360966584945360484972547, 4.44523923504762349000224992800, 5.34199039125556934636135680191, 6.44169037794429369403741529717, 6.89246668485072600270343561885, 7.74285862214245067124644104735
