| L(s) = 1 | − 53.2·3-s + 88.7·5-s − 1.09e3·7-s + 645.·9-s − 247.·11-s + 5.31e3·13-s − 4.72e3·15-s + 8.38e3·17-s − 6.85e3·19-s + 5.82e4·21-s + 1.18e3·23-s − 7.02e4·25-s + 8.20e4·27-s + 1.63e5·29-s − 2.87e4·31-s + 1.31e4·33-s − 9.71e4·35-s + 2.10e5·37-s − 2.83e5·39-s + 6.36e5·41-s − 3.01e5·43-s + 5.73e4·45-s − 2.95e5·47-s + 3.73e5·49-s − 4.46e5·51-s + 4.64e5·53-s − 2.20e4·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s + 0.317·5-s − 1.20·7-s + 0.295·9-s − 0.0561·11-s + 0.671·13-s − 0.361·15-s + 0.413·17-s − 0.229·19-s + 1.37·21-s + 0.0202·23-s − 0.899·25-s + 0.801·27-s + 1.24·29-s − 0.173·31-s + 0.0639·33-s − 0.382·35-s + 0.683·37-s − 0.764·39-s + 1.44·41-s − 0.578·43-s + 0.0938·45-s − 0.414·47-s + 0.453·49-s − 0.470·51-s + 0.428·53-s − 0.0178·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 + 53.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 88.7T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.09e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 247.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.31e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.38e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 1.18e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.63e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.87e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.36e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.64e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.16e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.70e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.62e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.42e6T + 8.07e13T^{2} \) |
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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
−10.13391782164916194107346074530, −9.331587963707626722105172962406, −8.098546769522139294562654760049, −6.68035477784931403085311810050, −6.14667553173674525951847749813, −5.29844916328595809263024542190, −3.94185780779164835239964158254, −2.70048375857876829386303417468, −1.03877455892235758701737169604, 0,
1.03877455892235758701737169604, 2.70048375857876829386303417468, 3.94185780779164835239964158254, 5.29844916328595809263024542190, 6.14667553173674525951847749813, 6.68035477784931403085311810050, 8.098546769522139294562654760049, 9.331587963707626722105172962406, 10.13391782164916194107346074530
