| L(s) = 1 | − 10.3·2-s − 20·4-s + 353.·5-s − 559·7-s + 1.53e3·8-s − 3.67e3·10-s − 4.71e3·11-s − 8.67e3·13-s + 5.80e3·14-s − 1.34e4·16-s − 2.51e4·17-s − 3.24e4·19-s − 7.06e3·20-s + 4.90e4·22-s + 8.24e4·23-s + 4.67e4·25-s + 9.01e4·26-s + 1.11e4·28-s − 1.57e5·29-s + 2.29e5·31-s − 5.73e4·32-s + 2.61e5·34-s − 1.97e5·35-s − 5.41e5·37-s + 3.37e5·38-s + 5.43e5·40-s − 3.53e5·41-s + ⋯ |
| L(s) = 1 | − 0.918·2-s − 0.156·4-s + 1.26·5-s − 0.615·7-s + 1.06·8-s − 1.16·10-s − 1.06·11-s − 1.09·13-s + 0.565·14-s − 0.819·16-s − 1.24·17-s − 1.08·19-s − 0.197·20-s + 0.981·22-s + 1.41·23-s + 0.598·25-s + 1.00·26-s + 0.0962·28-s − 1.20·29-s + 1.38·31-s − 0.309·32-s + 1.13·34-s − 0.778·35-s − 1.75·37-s + 0.997·38-s + 1.34·40-s − 0.801·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 10.3T + 128T^{2} \) |
| 5 | \( 1 - 353.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 559T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.71e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.51e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.24e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.41e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.85e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.37e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.14e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.97e6T + 8.07e13T^{2} \) |
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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
−15.30809859411365159597219379717, −13.62928834236558798218828006344, −12.92765914721750209441987748146, −10.62339328422532919484579588179, −9.777998979349594916686192167271, −8.687511773366402681957283490438, −6.90743720729900396756342078891, −5.05739494140163633204737767854, −2.21375110206006730763230573384, 0,
2.21375110206006730763230573384, 5.05739494140163633204737767854, 6.90743720729900396756342078891, 8.687511773366402681957283490438, 9.777998979349594916686192167271, 10.62339328422532919484579588179, 12.92765914721750209441987748146, 13.62928834236558798218828006344, 15.30809859411365159597219379717
