| L(s) = 1 | − 2-s − 1.41·3-s − 4-s + 1.41·5-s + 1.41·6-s + 3·8-s − 0.999·9-s − 1.41·10-s − 11-s + 1.41·12-s − 2.00·15-s − 16-s + 0.999·18-s + 5.65·19-s − 1.41·20-s + 22-s − 2·23-s − 4.24·24-s − 2.99·25-s + 5.65·27-s − 6·29-s + 2.00·30-s − 4.24·31-s − 5·32-s + 1.41·33-s + 0.999·36-s − 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.816·3-s − 0.5·4-s + 0.632·5-s + 0.577·6-s + 1.06·8-s − 0.333·9-s − 0.447·10-s − 0.301·11-s + 0.408·12-s − 0.516·15-s − 0.250·16-s + 0.235·18-s + 1.29·19-s − 0.316·20-s + 0.213·22-s − 0.417·23-s − 0.866·24-s − 0.599·25-s + 1.08·27-s − 1.11·29-s + 0.365·30-s − 0.762·31-s − 0.883·32-s + 0.246·33-s + 0.166·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−10.19529214284627067272331286002, −9.641428039501836477067445544568, −8.717716498944729327357489039624, −7.81880566865805636108283565064, −6.76307776275674724656177094063, −5.51877644588746354149892735408, −5.10543977750753996663514609695, −3.53315722473595459185823092430, −1.69743425806534260616991167717, 0,
1.69743425806534260616991167717, 3.53315722473595459185823092430, 5.10543977750753996663514609695, 5.51877644588746354149892735408, 6.76307776275674724656177094063, 7.81880566865805636108283565064, 8.717716498944729327357489039624, 9.641428039501836477067445544568, 10.19529214284627067272331286002
