| L(s) = 1 | + 76.3·2-s − 243·3-s + 3.78e3·4-s + 5.26e3·5-s − 1.85e4·6-s + 2.13e4·7-s + 1.32e5·8-s + 5.90e4·9-s + 4.01e5·10-s − 3.98e5·11-s − 9.18e5·12-s − 5.98e5·13-s + 1.62e6·14-s − 1.27e6·15-s + 2.35e6·16-s − 9.96e6·17-s + 4.50e6·18-s − 5.59e6·19-s + 1.98e7·20-s − 5.18e6·21-s − 3.04e7·22-s − 1.67e7·23-s − 3.21e7·24-s − 2.11e7·25-s − 4.57e7·26-s − 1.43e7·27-s + 8.05e7·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.84·4-s + 0.752·5-s − 0.974·6-s + 0.479·7-s + 1.42·8-s + 0.333·9-s + 1.27·10-s − 0.746·11-s − 1.06·12-s − 0.447·13-s + 0.808·14-s − 0.434·15-s + 0.561·16-s − 1.70·17-s + 0.562·18-s − 0.518·19-s + 1.39·20-s − 0.276·21-s − 1.25·22-s − 0.542·23-s − 0.824·24-s − 0.433·25-s − 0.754·26-s − 0.192·27-s + 0.885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 76.3T + 2.04e3T^{2} \) |
| 5 | \( 1 - 5.26e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.13e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.98e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 5.98e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.96e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 5.59e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.67e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 8.80e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 5.31e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.54e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.12e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 7.10e7T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.50e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.65e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.70e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.45e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.40e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.37e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.87e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.54e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.57e11T + 7.15e21T^{2} \) |
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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
−10.69551219936795671485366101097, −9.358635768490403029752822724357, −7.72796700482015958471985738292, −6.51007123307516007048553349645, −5.81645189404258429236819965428, −4.87588441004005852911110920883, −4.16063798607595002822066813191, −2.56833831714961643591436991959, −1.86888838452920132534993653762, 0,
1.86888838452920132534993653762, 2.56833831714961643591436991959, 4.16063798607595002822066813191, 4.87588441004005852911110920883, 5.81645189404258429236819965428, 6.51007123307516007048553349645, 7.72796700482015958471985738292, 9.358635768490403029752822724357, 10.69551219936795671485366101097
