| L(s) = 1 | − 3.18·2-s − 117.·4-s + 0.588·5-s + 783.·8-s − 1.87·10-s + 1.19e3·11-s + 7.65e3·13-s + 1.25e4·16-s − 3.51e4·17-s − 8.57e3·19-s − 69.4·20-s − 3.82e3·22-s − 7.87e4·23-s − 7.81e4·25-s − 2.43e4·26-s + 1.99e5·29-s + 1.37e5·31-s − 1.40e5·32-s + 1.12e5·34-s + 9.00e4·37-s + 2.73e4·38-s + 461.·40-s + 2.69e5·41-s + 6.02e5·43-s − 1.41e5·44-s + 2.50e5·46-s + 3.84e5·47-s + ⋯ |
| L(s) = 1 | − 0.281·2-s − 0.920·4-s + 0.00210·5-s + 0.540·8-s − 0.000593·10-s + 0.271·11-s + 0.965·13-s + 0.768·16-s − 1.73·17-s − 0.286·19-s − 0.00194·20-s − 0.0765·22-s − 1.34·23-s − 0.999·25-s − 0.271·26-s + 1.52·29-s + 0.826·31-s − 0.757·32-s + 0.489·34-s + 0.292·37-s + 0.0807·38-s + 0.00113·40-s + 0.610·41-s + 1.15·43-s − 0.250·44-s + 0.379·46-s + 0.540·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.18T + 128T^{2} \) |
| 5 | \( 1 - 0.588T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.19e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.65e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.51e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.57e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.87e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.99e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.00e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.69e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.02e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.84e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.64e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.99e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.37e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.63e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.42e6T + 8.07e13T^{2} \) |
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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
−9.425287730166470657339141505930, −8.586679610951282419413898181870, −8.016308538294000740722753658337, −6.64788275573201067224946040617, −5.80065730461907802958066541318, −4.44011476455477270870109067704, −3.94639904877733401403592233090, −2.35755624612048212296137938252, −1.07011418148179243069892446338, 0,
1.07011418148179243069892446338, 2.35755624612048212296137938252, 3.94639904877733401403592233090, 4.44011476455477270870109067704, 5.80065730461907802958066541318, 6.64788275573201067224946040617, 8.016308538294000740722753658337, 8.586679610951282419413898181870, 9.425287730166470657339141505930
