| L(s) = 1 | − 213.·2-s − 6.56e3·3-s − 8.52e4·4-s − 3.90e5·5-s + 1.40e6·6-s + 7.24e6·7-s + 4.62e7·8-s + 4.30e7·9-s + 8.35e7·10-s + 4.10e7·11-s + 5.59e8·12-s + 1.35e9·13-s − 1.54e9·14-s + 2.56e9·15-s + 1.27e9·16-s + 1.84e10·17-s − 9.21e9·18-s + 1.42e9·19-s + 3.33e10·20-s − 4.75e10·21-s − 8.78e9·22-s − 1.96e10·23-s − 3.03e11·24-s + 1.52e11·25-s − 2.90e11·26-s − 2.82e11·27-s − 6.17e11·28-s + ⋯ |
| L(s) = 1 | − 0.590·2-s − 0.577·3-s − 0.650·4-s − 0.447·5-s + 0.341·6-s + 0.474·7-s + 0.975·8-s + 0.333·9-s + 0.264·10-s + 0.0577·11-s + 0.375·12-s + 0.461·13-s − 0.280·14-s + 0.258·15-s + 0.0742·16-s + 0.639·17-s − 0.196·18-s + 0.0192·19-s + 0.291·20-s − 0.274·21-s − 0.0341·22-s − 0.0522·23-s − 0.563·24-s + 0.200·25-s − 0.272·26-s − 0.192·27-s − 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 + 213.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 7.24e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 4.10e7T + 5.05e17T^{2} \) |
| 13 | \( 1 - 1.35e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.84e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.42e9T + 5.48e21T^{2} \) |
| 23 | \( 1 + 1.96e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.22e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.93e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.47e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.91e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 9.80e12T + 5.87e27T^{2} \) |
| 47 | \( 1 - 6.08e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.72e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.78e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.98e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.22e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.65e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.26e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.63e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.45e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.52e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.59e16T + 5.95e33T^{2} \) |
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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
−14.53466780676039170730011887457, −13.04269245413584692987990035119, −11.52850479255845904371537633625, −10.24248481156160007197878960126, −8.734585716711850213088899101146, −7.42670895449038680973180058743, −5.39110963581099086175084976020, −3.94772994680664447512180419953, −1.34401625379564660685743950964, 0,
1.34401625379564660685743950964, 3.94772994680664447512180419953, 5.39110963581099086175084976020, 7.42670895449038680973180058743, 8.734585716711850213088899101146, 10.24248481156160007197878960126, 11.52850479255845904371537633625, 13.04269245413584692987990035119, 14.53466780676039170730011887457
