| L(s) = 1 | − 16·2-s + 256·4-s + 1.11e3·5-s − 4.30e3·7-s − 4.09e3·8-s − 1.79e4·10-s − 1.44e4·11-s + 1.20e5·13-s + 6.88e4·14-s + 6.55e4·16-s − 9.99e3·17-s + 9.94e4·19-s + 2.86e5·20-s + 2.31e5·22-s − 2.12e6·23-s − 6.99e5·25-s − 1.93e6·26-s − 1.10e6·28-s + 4.24e6·29-s − 9.72e6·31-s − 1.04e6·32-s + 1.59e5·34-s − 4.81e6·35-s + 1.35e7·37-s − 1.59e6·38-s − 4.58e6·40-s + 2.10e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.801·5-s − 0.677·7-s − 0.353·8-s − 0.566·10-s − 0.297·11-s + 1.17·13-s + 0.478·14-s + 0.250·16-s − 0.0290·17-s + 0.175·19-s + 0.400·20-s + 0.210·22-s − 1.58·23-s − 0.358·25-s − 0.829·26-s − 0.338·28-s + 1.11·29-s − 1.89·31-s − 0.176·32-s + 0.0205·34-s − 0.542·35-s + 1.19·37-s − 0.123·38-s − 0.283·40-s + 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.11e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.20e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.99e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.94e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.24e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.35e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.93e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.79e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.61e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.55e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.91e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.22e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.93e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.31e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.14e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.26e8T + 7.60e17T^{2} \) |
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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
−10.41301436869588671757114367167, −9.682560986310483570149223476985, −8.766141888696555595674061667227, −7.66184046684297064038134286939, −6.35295159490772670615737474479, −5.70974106940730794741048160202, −3.85657525780767177891181305527, −2.51014873033431379299000991386, −1.35977700892932588490247398790, 0,
1.35977700892932588490247398790, 2.51014873033431379299000991386, 3.85657525780767177891181305527, 5.70974106940730794741048160202, 6.35295159490772670615737474479, 7.66184046684297064038134286939, 8.766141888696555595674061667227, 9.682560986310483570149223476985, 10.41301436869588671757114367167
