| L(s) = 1 | + 16.7·2-s + 151.·4-s + 298.·5-s + 200.·7-s + 395.·8-s + 4.99e3·10-s − 6.81e3·11-s − 7.65e3·13-s + 3.34e3·14-s − 1.27e4·16-s + 2.38e4·17-s + 4.14e4·19-s + 4.53e4·20-s − 1.14e5·22-s − 8.70e4·23-s + 1.11e4·25-s − 1.27e5·26-s + 3.03e4·28-s + 2.88e4·29-s − 1.36e5·31-s − 2.64e5·32-s + 3.99e5·34-s + 5.97e4·35-s − 2.38e5·37-s + 6.93e5·38-s + 1.18e5·40-s + 3.87e5·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.18·4-s + 1.06·5-s + 0.220·7-s + 0.273·8-s + 1.58·10-s − 1.54·11-s − 0.966·13-s + 0.325·14-s − 0.780·16-s + 1.17·17-s + 1.38·19-s + 1.26·20-s − 2.28·22-s − 1.49·23-s + 0.142·25-s − 1.42·26-s + 0.261·28-s + 0.219·29-s − 0.820·31-s − 1.42·32-s + 1.74·34-s + 0.235·35-s − 0.773·37-s + 2.04·38-s + 0.292·40-s + 0.877·41-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 | \( 1 - 16.7T + 128T^{2} \) |
| 5 | \( 1 - 298.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 200.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.65e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.88e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.60e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.90e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.23e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.27e7T + 8.07e13T^{2} \) |
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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
−9.699669755002502128901861951415, −8.102099874131583944313349498910, −7.27744332983439445942744089770, −6.03055035905811446243270425096, −5.35880618244941438207124839233, −4.92604601782355904252217483487, −3.48837299515879501584228332265, −2.63342835226359266991216686156, −1.75261392770753340049451716395, 0,
1.75261392770753340049451716395, 2.63342835226359266991216686156, 3.48837299515879501584228332265, 4.92604601782355904252217483487, 5.35880618244941438207124839233, 6.03055035905811446243270425096, 7.27744332983439445942744089770, 8.102099874131583944313349498910, 9.699669755002502128901861951415
