| L(s) = 1 | − 3.27e4·2-s + 2.09e7·3-s + 1.07e9·4-s + 3.05e10·5-s − 6.85e11·6-s + 1.72e13·7-s − 3.51e13·8-s − 1.80e14·9-s − 1.00e15·10-s − 1.40e16·11-s + 2.24e16·12-s + 1.12e17·13-s − 5.66e17·14-s + 6.38e17·15-s + 1.15e18·16-s − 1.87e19·17-s + 5.91e18·18-s − 1.17e20·19-s + 3.27e19·20-s + 3.61e20·21-s + 4.60e20·22-s − 2.47e21·23-s − 7.35e20·24-s + 9.31e20·25-s − 3.67e21·26-s − 1.66e22·27-s + 1.85e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.841·3-s + 0.5·4-s + 0.447·5-s − 0.594·6-s + 1.37·7-s − 0.353·8-s − 0.292·9-s − 0.316·10-s − 1.01·11-s + 0.420·12-s + 0.607·13-s − 0.973·14-s + 0.376·15-s + 0.250·16-s − 1.59·17-s + 0.206·18-s − 1.77·19-s + 0.223·20-s + 1.15·21-s + 0.716·22-s − 1.93·23-s − 0.297·24-s + 0.200·25-s − 0.429·26-s − 1.08·27-s + 0.688·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| 5 | \( 1 - 3.05e10T \) |
| good | 3 | \( 1 - 2.09e7T + 6.17e14T^{2} \) |
| 7 | \( 1 - 1.72e13T + 1.57e26T^{2} \) |
| 11 | \( 1 + 1.40e16T + 1.91e32T^{2} \) |
| 13 | \( 1 - 1.12e17T + 3.40e34T^{2} \) |
| 17 | \( 1 + 1.87e19T + 1.39e38T^{2} \) |
| 19 | \( 1 + 1.17e20T + 4.37e39T^{2} \) |
| 23 | \( 1 + 2.47e21T + 1.63e42T^{2} \) |
| 29 | \( 1 - 6.20e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 1.42e23T + 1.70e46T^{2} \) |
| 37 | \( 1 - 1.53e24T + 4.11e48T^{2} \) |
| 41 | \( 1 + 6.64e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 1.23e24T + 4.34e50T^{2} \) |
| 47 | \( 1 + 1.65e26T + 6.83e51T^{2} \) |
| 53 | \( 1 - 2.10e25T + 2.83e53T^{2} \) |
| 59 | \( 1 - 2.11e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 2.54e27T + 2.21e55T^{2} \) |
| 67 | \( 1 - 2.58e28T + 4.05e56T^{2} \) |
| 71 | \( 1 + 6.76e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 5.88e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 2.13e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 1.06e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 5.07e29T + 2.69e60T^{2} \) |
| 97 | \( 1 - 2.99e30T + 3.88e61T^{2} \) |
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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
−13.36071533348091776094050780466, −11.40245634256205334941756820671, −10.26924689275884396820989377421, −8.481301101531445725476991858886, −8.213945457805781123827835595005, −6.25248415101165197798687948959, −4.49057038408816486893496607961, −2.50451788042048994607758862042, −1.81074769645060792390397805515, 0,
1.81074769645060792390397805515, 2.50451788042048994607758862042, 4.49057038408816486893496607961, 6.25248415101165197798687948959, 8.213945457805781123827835595005, 8.481301101531445725476991858886, 10.26924689275884396820989377421, 11.40245634256205334941756820671, 13.36071533348091776094050780466
