| L(s) = 1 | + 8·2-s + 20.8·3-s + 64·4-s − 216.·5-s + 167.·6-s + 512·8-s − 1.75e3·9-s − 1.73e3·10-s + 2.58e3·11-s + 1.33e3·12-s − 8.92e3·13-s − 4.52e3·15-s + 4.09e3·16-s − 1.11e4·17-s − 1.40e4·18-s − 8.64e3·19-s − 1.38e4·20-s + 2.06e4·22-s − 6.60e4·23-s + 1.06e4·24-s − 3.11e4·25-s − 7.13e4·26-s − 8.22e4·27-s + 1.28e5·29-s − 3.62e4·30-s − 2.04e5·31-s + 3.27e4·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.446·3-s + 0.5·4-s − 0.775·5-s + 0.315·6-s + 0.353·8-s − 0.800·9-s − 0.548·10-s + 0.584·11-s + 0.223·12-s − 1.12·13-s − 0.346·15-s + 0.250·16-s − 0.548·17-s − 0.566·18-s − 0.289·19-s − 0.387·20-s + 0.413·22-s − 1.13·23-s + 0.157·24-s − 0.398·25-s − 0.796·26-s − 0.803·27-s + 0.980·29-s − 0.244·30-s − 1.23·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 20.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 216.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.58e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.64e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.91e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.27e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.68e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.12e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.39e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.51e6T + 8.07e13T^{2} \) |
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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
−11.94952056425295100326996705992, −11.33614252973948343178998940169, −9.830347172525027233888824088551, −8.484048284777853645401535499316, −7.47588816077862334194820516787, −6.15583689703063279310621329978, −4.66056412807560868795343194097, −3.51789900430071247710535886748, −2.20652309718507945842168467947, 0,
2.20652309718507945842168467947, 3.51789900430071247710535886748, 4.66056412807560868795343194097, 6.15583689703063279310621329978, 7.47588816077862334194820516787, 8.484048284777853645401535499316, 9.830347172525027233888824088551, 11.33614252973948343178998940169, 11.94952056425295100326996705992
