| L(s) = 1 | − 13.4·2-s − 87.0·3-s + 53.3·4-s + 131.·5-s + 1.17e3·6-s − 712.·7-s + 1.00e3·8-s + 5.39e3·9-s − 1.76e3·10-s + 6.10e3·11-s − 4.64e3·12-s − 7.92e3·13-s + 9.59e3·14-s − 1.14e4·15-s − 2.03e4·16-s − 1.63e4·17-s − 7.26e4·18-s + 3.95e4·19-s + 6.99e3·20-s + 6.20e4·21-s − 8.21e4·22-s + 2.92e4·23-s − 8.75e4·24-s − 6.09e4·25-s + 1.06e5·26-s − 2.79e5·27-s − 3.80e4·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 1.86·3-s + 0.416·4-s + 0.469·5-s + 2.21·6-s − 0.784·7-s + 0.694·8-s + 2.46·9-s − 0.558·10-s + 1.38·11-s − 0.776·12-s − 1.00·13-s + 0.934·14-s − 0.873·15-s − 1.24·16-s − 0.806·17-s − 2.93·18-s + 1.32·19-s + 0.195·20-s + 1.46·21-s − 1.64·22-s + 0.502·23-s − 1.29·24-s − 0.779·25-s + 1.19·26-s − 2.72·27-s − 0.327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 13.4T + 128T^{2} \) |
| 3 | \( 1 + 87.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 131.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 712.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.10e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.95e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.92e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.11e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.68e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 9.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.77e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.31e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.10e4T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.31e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.08e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.22e6T + 8.07e13T^{2} \) |
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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
−13.61857808526654266706698422318, −12.23045937234941381291613579379, −11.28329494506955011585615621564, −9.992379285979658988448159923362, −9.422892785887211889315585876188, −7.19280931221569374200915913394, −6.23772209148749328684189776479, −4.65801383280346497820947174918, −1.26161444862495668862143372586, 0,
1.26161444862495668862143372586, 4.65801383280346497820947174918, 6.23772209148749328684189776479, 7.19280931221569374200915913394, 9.422892785887211889315585876188, 9.992379285979658988448159923362, 11.28329494506955011585615621564, 12.23045937234941381291613579379, 13.61857808526654266706698422318
