| L(s) = 1 | + 223.·5-s + 1.03e3·7-s − 4.73e3·11-s + 9.83e3·13-s − 2.78e4·17-s − 3.33e4·19-s − 2.86e4·23-s − 2.83e4·25-s + 1.27e4·29-s − 8.51e4·31-s + 2.31e5·35-s − 4.39e5·37-s − 8.67e5·41-s + 2.58e5·43-s + 4.89e5·47-s + 2.50e5·49-s + 1.08e6·53-s − 1.05e6·55-s − 4.90e5·59-s + 1.88e6·61-s + 2.19e6·65-s + 4.60e6·67-s − 1.31e6·71-s − 4.44e6·73-s − 4.90e6·77-s + 1.07e6·79-s − 1.01e7·83-s + ⋯ |
| L(s) = 1 | + 0.797·5-s + 1.14·7-s − 1.07·11-s + 1.24·13-s − 1.37·17-s − 1.11·19-s − 0.490·23-s − 0.363·25-s + 0.0971·29-s − 0.513·31-s + 0.911·35-s − 1.42·37-s − 1.96·41-s + 0.494·43-s + 0.688·47-s + 0.303·49-s + 1.00·53-s − 0.856·55-s − 0.310·59-s + 1.06·61-s + 0.990·65-s + 1.87·67-s − 0.434·71-s − 1.33·73-s − 1.22·77-s + 0.244·79-s − 1.94·83-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 | \( 1 - 223.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.03e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.73e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.83e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.78e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.33e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.27e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 8.51e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.67e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.08e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.90e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.88e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.60e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.44e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.01e7T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.33e6T + 8.07e13T^{2} \) |
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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
−10.15469092531165288267111004743, −8.714799858385792666988625333965, −8.340306869380536609897659831005, −6.99761334544464260694886581917, −5.92360440493614150115299824106, −5.05553770344955219248325994329, −3.94724540962778523958731590458, −2.30596781207506100862531197702, −1.61880671532859283688459051685, 0,
1.61880671532859283688459051685, 2.30596781207506100862531197702, 3.94724540962778523958731590458, 5.05553770344955219248325994329, 5.92360440493614150115299824106, 6.99761334544464260694886581917, 8.340306869380536609897659831005, 8.714799858385792666988625333965, 10.15469092531165288267111004743
