| L(s) = 1 | + 68.3·3-s − 180.·5-s + 343·7-s + 2.48e3·9-s − 392.·11-s − 6.22e3·13-s − 1.23e4·15-s − 1.81e3·17-s + 5.49e4·19-s + 2.34e4·21-s − 1.00e5·23-s − 4.55e4·25-s + 2.02e4·27-s − 4.67e4·29-s + 2.20e4·31-s − 2.68e4·33-s − 6.19e4·35-s + 1.80e5·37-s − 4.25e5·39-s − 9.16e3·41-s − 1.73e5·43-s − 4.48e5·45-s − 1.04e6·47-s + 1.17e5·49-s − 1.24e5·51-s + 1.25e6·53-s + 7.09e4·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s − 0.646·5-s + 0.377·7-s + 1.13·9-s − 0.0889·11-s − 0.785·13-s − 0.944·15-s − 0.0897·17-s + 1.83·19-s + 0.552·21-s − 1.72·23-s − 0.582·25-s + 0.197·27-s − 0.355·29-s + 0.133·31-s − 0.130·33-s − 0.244·35-s + 0.584·37-s − 1.14·39-s − 0.0207·41-s − 0.332·43-s − 0.733·45-s − 1.47·47-s + 0.142·49-s − 0.131·51-s + 1.15·53-s + 0.0574·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 68.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 180.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 392.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.22e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.81e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.00e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.67e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.20e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 9.16e3T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.25e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.44e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.91e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.86e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.22e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
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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
−9.527924815339979025263510057128, −8.357264657295481556395139323056, −7.83496484038830031850656995848, −7.18397951309457562776671464735, −5.61404892767704206772143130248, −4.37781883405165493663106518607, −3.51039968371510075211036954071, −2.58614121186491720094226173676, −1.54831940492250953697221899594, 0,
1.54831940492250953697221899594, 2.58614121186491720094226173676, 3.51039968371510075211036954071, 4.37781883405165493663106518607, 5.61404892767704206772143130248, 7.18397951309457562776671464735, 7.83496484038830031850656995848, 8.357264657295481556395139323056, 9.527924815339979025263510057128
