| L(s) = 1 | + 7.24·2-s − 75.4·4-s − 1.17·5-s + 1.09e3·7-s − 1.47e3·8-s − 8.48·10-s + 7.87e3·11-s − 7.98e3·13-s + 7.95e3·14-s − 1.02e3·16-s − 1.48e4·17-s − 3.72e4·19-s + 88.3·20-s + 5.71e4·22-s + 4.96e4·23-s − 7.81e4·25-s − 5.78e4·26-s − 8.28e4·28-s − 1.43e5·29-s + 1.94e5·31-s + 1.81e5·32-s − 1.07e5·34-s − 1.28e3·35-s − 1.19e5·37-s − 2.70e5·38-s + 1.72e3·40-s − 6.18e5·41-s + ⋯ |
| L(s) = 1 | + 0.640·2-s − 0.589·4-s − 0.00418·5-s + 1.20·7-s − 1.01·8-s − 0.00268·10-s + 1.78·11-s − 1.00·13-s + 0.774·14-s − 0.0626·16-s − 0.731·17-s − 1.24·19-s + 0.00246·20-s + 1.14·22-s + 0.850·23-s − 0.999·25-s − 0.645·26-s − 0.713·28-s − 1.09·29-s + 1.17·31-s + 0.978·32-s − 0.468·34-s − 0.00506·35-s − 0.388·37-s − 0.798·38-s + 0.00426·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 7.24T + 128T^{2} \) |
| 5 | \( 1 + 1.17T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.09e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.87e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.98e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.72e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.94e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.19e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.58e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.94e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.11e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.46e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.98e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.17e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.35e7T + 8.07e13T^{2} \) |
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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
−9.030658894851194970069727056269, −8.630762489185171105195689107114, −7.38418692116646006036988348124, −6.38445786935247811595034493061, −5.35890221626483169203086229248, −4.38513611839594520974759783702, −3.98551560126463873532002388104, −2.42707214026008109851638015300, −1.31596174688188425952410960406, 0,
1.31596174688188425952410960406, 2.42707214026008109851638015300, 3.98551560126463873532002388104, 4.38513611839594520974759783702, 5.35890221626483169203086229248, 6.38445786935247811595034493061, 7.38418692116646006036988348124, 8.630762489185171105195689107114, 9.030658894851194970069727056269
