| L(s) = 1 | − 16.8·2-s + 109.·3-s − 226.·4-s − 2.43e3·5-s − 1.85e3·6-s − 2.40e3·7-s + 1.24e4·8-s − 7.62e3·9-s + 4.11e4·10-s − 2.85e4·11-s − 2.48e4·12-s + 1.38e5·13-s + 4.05e4·14-s − 2.67e5·15-s − 9.47e4·16-s − 1.01e5·17-s + 1.28e5·18-s − 4.88e5·19-s + 5.52e5·20-s − 2.63e5·21-s + 4.82e5·22-s − 1.40e5·23-s + 1.37e6·24-s + 3.99e6·25-s − 2.33e6·26-s − 2.99e6·27-s + 5.44e5·28-s + ⋯ |
| L(s) = 1 | − 0.746·2-s + 0.782·3-s − 0.442·4-s − 1.74·5-s − 0.584·6-s − 0.377·7-s + 1.07·8-s − 0.387·9-s + 1.30·10-s − 0.587·11-s − 0.346·12-s + 1.34·13-s + 0.282·14-s − 1.36·15-s − 0.361·16-s − 0.293·17-s + 0.289·18-s − 0.860·19-s + 0.772·20-s − 0.295·21-s + 0.438·22-s − 0.104·23-s + 0.843·24-s + 2.04·25-s − 1.00·26-s − 1.08·27-s + 0.167·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 16.8T + 512T^{2} \) |
| 3 | \( 1 - 109.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.43e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.38e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.01e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.40e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.19e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.65e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.67e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.81e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.50e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.18e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.89e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.68e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.75e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.36e8T + 7.60e17T^{2} \) |
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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
−19.40675896893413344248306842164, −18.50866241667015281535077673490, −16.45240372921716870509386384917, −15.08781012950683421113327219960, −13.22126393463707655086164512156, −11.05889941982065285218625329280, −8.815004289195965606401040694219, −7.85307835004733678158176389773, −3.79837678420596625160071144778, 0,
3.79837678420596625160071144778, 7.85307835004733678158176389773, 8.815004289195965606401040694219, 11.05889941982065285218625329280, 13.22126393463707655086164512156, 15.08781012950683421113327219960, 16.45240372921716870509386384917, 18.50866241667015281535077673490, 19.40675896893413344248306842164
