| L(s) = 1 | − 9.74·2-s − 33.0·4-s − 321.·5-s − 939.·7-s + 1.56e3·8-s + 3.13e3·10-s − 4.99e3·11-s − 1.23e3·13-s + 9.15e3·14-s − 1.10e4·16-s − 9.78e3·17-s + 1.93e4·19-s + 1.06e4·20-s + 4.87e4·22-s − 7.98e4·23-s + 2.54e4·25-s + 1.20e4·26-s + 3.10e4·28-s − 1.44e5·29-s + 1.48e5·31-s − 9.31e4·32-s + 9.53e4·34-s + 3.02e5·35-s + 2.03e5·37-s − 1.88e5·38-s − 5.05e5·40-s + 3.93e5·41-s + ⋯ |
| L(s) = 1 | − 0.861·2-s − 0.258·4-s − 1.15·5-s − 1.03·7-s + 1.08·8-s + 0.991·10-s − 1.13·11-s − 0.155·13-s + 0.892·14-s − 0.675·16-s − 0.482·17-s + 0.646·19-s + 0.297·20-s + 0.975·22-s − 1.36·23-s + 0.326·25-s + 0.134·26-s + 0.267·28-s − 1.10·29-s + 0.894·31-s − 0.502·32-s + 0.415·34-s + 1.19·35-s + 0.661·37-s − 0.556·38-s − 1.24·40-s + 0.890·41-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 | \( 1 + 9.74T + 128T^{2} \) |
| 5 | \( 1 + 321.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 939.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 9.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.93e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.48e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.03e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.43e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.86e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.28e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.54e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.62e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.86e6T + 8.07e13T^{2} \) |
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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
−9.384620633249206025651840587410, −8.176171149238924212322181647417, −7.82228033228614944012178700583, −6.87668483691020059381852626169, −5.57374566138870355625817665932, −4.39121574249240535534043334640, −3.57029593399697294845135204940, −2.31886158851311264679102610149, −0.66513323452501202392668785481, 0,
0.66513323452501202392668785481, 2.31886158851311264679102610149, 3.57029593399697294845135204940, 4.39121574249240535534043334640, 5.57374566138870355625817665932, 6.87668483691020059381852626169, 7.82228033228614944012178700583, 8.176171149238924212322181647417, 9.384620633249206025651840587410
