| L(s) = 1 | + 346.·2-s + 6.56e3·3-s − 1.10e4·4-s + 3.90e5·5-s + 2.27e6·6-s − 1.80e7·7-s − 4.92e7·8-s + 4.30e7·9-s + 1.35e8·10-s − 1.45e8·11-s − 7.23e7·12-s − 3.26e9·13-s − 6.26e9·14-s + 2.56e9·15-s − 1.56e10·16-s + 7.81e9·17-s + 1.49e10·18-s − 7.18e10·19-s − 4.30e9·20-s − 1.18e11·21-s − 5.03e10·22-s + 1.67e11·23-s − 3.23e11·24-s + 1.52e11·25-s − 1.13e12·26-s + 2.82e11·27-s + 1.99e11·28-s + ⋯ |
| L(s) = 1 | + 0.957·2-s + 0.577·3-s − 0.0841·4-s + 0.447·5-s + 0.552·6-s − 1.18·7-s − 1.03·8-s + 0.333·9-s + 0.427·10-s − 0.204·11-s − 0.0485·12-s − 1.10·13-s − 1.13·14-s + 0.258·15-s − 0.908·16-s + 0.271·17-s + 0.319·18-s − 0.970·19-s − 0.0376·20-s − 0.684·21-s − 0.195·22-s + 0.445·23-s − 0.599·24-s + 0.200·25-s − 1.06·26-s + 0.192·27-s + 0.0997·28-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 | \( 1 - 346.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 1.80e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.45e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.26e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 7.81e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 7.18e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.67e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.63e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.38e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.68e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 7.52e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.73e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.42e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 8.01e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 9.64e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.29e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.71e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.29e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 6.86e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.21e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.23e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.91e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.26e17T + 5.95e33T^{2} \) |
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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
−14.32572744533670611798593419045, −13.17906543526777722599927347812, −12.40162042529557743808462913412, −10.06848893939569723753244720963, −8.924554600898746256317057745820, −6.82218911287130375549995724698, −5.29767319805013976579092034304, −3.69708035424296950539879664778, −2.47722345968304888298274321211, 0,
2.47722345968304888298274321211, 3.69708035424296950539879664778, 5.29767319805013976579092034304, 6.82218911287130375549995724698, 8.924554600898746256317057745820, 10.06848893939569723753244720963, 12.40162042529557743808462913412, 13.17906543526777722599927347812, 14.32572744533670611798593419045
