| L(s) = 1 | + 11.8·2-s − 27·3-s + 11.6·4-s − 506.·5-s − 319.·6-s + 343·7-s − 1.37e3·8-s + 729·9-s − 5.98e3·10-s + 2.55e3·11-s − 314.·12-s − 845.·13-s + 4.05e3·14-s + 1.36e4·15-s − 1.77e4·16-s − 2.33e4·17-s + 8.61e3·18-s − 1.65e4·19-s − 5.89e3·20-s − 9.26e3·21-s + 3.02e4·22-s + 5.84e4·23-s + 3.71e4·24-s + 1.78e5·25-s − 9.98e3·26-s − 1.96e4·27-s + 3.99e3·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.577·3-s + 0.0909·4-s − 1.81·5-s − 0.603·6-s + 0.377·7-s − 0.949·8-s + 0.333·9-s − 1.89·10-s + 0.579·11-s − 0.0525·12-s − 0.106·13-s + 0.394·14-s + 1.04·15-s − 1.08·16-s − 1.15·17-s + 0.348·18-s − 0.554·19-s − 0.164·20-s − 0.218·21-s + 0.605·22-s + 1.00·23-s + 0.548·24-s + 2.28·25-s − 0.111·26-s − 0.192·27-s + 0.0343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 27T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 - 11.8T + 128T^{2} \) |
| 5 | \( 1 + 506.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 845.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.33e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.84e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.51e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.57e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.30e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.05e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.53e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.20e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 6.45e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.84e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.50e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.07e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.83e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.30e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.01e7T + 8.07e13T^{2} \) |
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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
−15.49353920862973173186092969531, −14.85134476712614553392459284003, −13.11353841118754649164283965869, −11.93048316411985654992076868186, −11.19125360556158486390801704942, −8.662300618383784066586513339856, −6.88728593278890824351320425883, −4.84819502217452942086127117110, −3.76338125254263880029545162404, 0,
3.76338125254263880029545162404, 4.84819502217452942086127117110, 6.88728593278890824351320425883, 8.662300618383784066586513339856, 11.19125360556158486390801704942, 11.93048316411985654992076868186, 13.11353841118754649164283965869, 14.85134476712614553392459284003, 15.49353920862973173186092969531
