| L(s) = 1 | − 8·2-s − 63.0·3-s + 64·4-s − 242.·5-s + 504.·6-s − 1.04e3·7-s − 512·8-s + 1.78e3·9-s + 1.94e3·10-s − 776.·11-s − 4.03e3·12-s + 4.34e3·13-s + 8.36e3·14-s + 1.52e4·15-s + 4.09e3·16-s − 2.12e4·17-s − 1.42e4·18-s + 4.05e4·19-s − 1.55e4·20-s + 6.59e4·21-s + 6.21e3·22-s + 3.43e4·23-s + 3.22e4·24-s − 1.92e4·25-s − 3.47e4·26-s + 2.52e4·27-s − 6.69e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 0.5·4-s − 0.867·5-s + 0.953·6-s − 1.15·7-s − 0.353·8-s + 0.816·9-s + 0.613·10-s − 0.175·11-s − 0.673·12-s + 0.548·13-s + 0.814·14-s + 1.16·15-s + 0.250·16-s − 1.04·17-s − 0.577·18-s + 1.35·19-s − 0.433·20-s + 1.55·21-s + 0.124·22-s + 0.588·23-s + 0.476·24-s − 0.247·25-s − 0.387·26-s + 0.247·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.007801785455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007801785455\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 + 63.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 242.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.04e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 776.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.34e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.05e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.56e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.66e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.11e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.94e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.62e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.16e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.25e7T + 8.07e13T^{2} \) |
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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
−9.660753254521800594431444760884, −9.010366219169969683029812321146, −7.71364613676056591524906342012, −6.98722602902404824026102473183, −6.15394721524736625554166896495, −5.33681529668123501553140755655, −4.00505048076096530370152087348, −2.99047169705939724383023212430, −1.31952676956878700277165460254, −0.04945787300275544121812853989,
0.04945787300275544121812853989, 1.31952676956878700277165460254, 2.99047169705939724383023212430, 4.00505048076096530370152087348, 5.33681529668123501553140755655, 6.15394721524736625554166896495, 6.98722602902404824026102473183, 7.71364613676056591524906342012, 9.010366219169969683029812321146, 9.660753254521800594431444760884
