| L(s) = 1 | + 114.·5-s − 1.47e3·7-s − 352.·11-s − 112.·13-s + 3.22e4·17-s + 1.70e4·19-s + 3.69e4·23-s − 6.49e4·25-s + 1.83e5·29-s + 7.93e4·31-s − 1.69e5·35-s − 5.10e5·37-s − 4.44e5·41-s − 7.19e4·43-s + 7.53e5·47-s + 1.34e6·49-s − 2.03e6·53-s − 4.04e4·55-s − 2.17e6·59-s − 2.03e6·61-s − 1.29e4·65-s + 1.72e6·67-s + 3.11e6·71-s − 7.30e5·73-s + 5.19e5·77-s + 1.08e6·79-s + 5.65e6·83-s + ⋯ |
| L(s) = 1 | + 0.411·5-s − 1.62·7-s − 0.0798·11-s − 0.0141·13-s + 1.59·17-s + 0.571·19-s + 0.633·23-s − 0.830·25-s + 1.39·29-s + 0.478·31-s − 0.667·35-s − 1.65·37-s − 1.00·41-s − 0.137·43-s + 1.05·47-s + 1.63·49-s − 1.87·53-s − 0.0328·55-s − 1.37·59-s − 1.14·61-s − 0.00582·65-s + 0.699·67-s + 1.03·71-s − 0.219·73-s + 0.129·77-s + 0.247·79-s + 1.08·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 114.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.47e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 352.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 112.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.22e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.70e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.69e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.83e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.93e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.19e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.53e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.17e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.03e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.11e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.30e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.10e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.45e6T + 8.07e13T^{2} \) |
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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
−9.880801568404594113333150841076, −9.275461726081917656684094688511, −8.019580321177448043173239357957, −6.90111463460344138932589157295, −6.09663470971270247213695879694, −5.12585050481438883863200504195, −3.54058319309912023497367956376, −2.86224313390557568919578566309, −1.26832673896512344676996807191, 0,
1.26832673896512344676996807191, 2.86224313390557568919578566309, 3.54058319309912023497367956376, 5.12585050481438883863200504195, 6.09663470971270247213695879694, 6.90111463460344138932589157295, 8.019580321177448043173239357957, 9.275461726081917656684094688511, 9.880801568404594113333150841076
