| L(s) = 1 | + 263.·5-s + 170.·7-s + 2.00e3·11-s − 1.27e4·13-s + 2.00e4·17-s + 6.77e3·19-s − 5.94e4·23-s − 8.81e3·25-s − 1.32e5·29-s − 1.87e5·31-s + 4.48e4·35-s − 3.32e5·37-s + 5.22e5·41-s + 4.52e5·43-s − 6.62e5·47-s − 7.94e5·49-s + 1.80e6·53-s + 5.27e5·55-s − 2.07e6·59-s + 1.99e6·61-s − 3.35e6·65-s − 3.12e6·67-s + 5.36e5·71-s + 2.35e6·73-s + 3.40e5·77-s + 5.73e6·79-s − 9.11e4·83-s + ⋯ |
| L(s) = 1 | + 0.941·5-s + 0.187·7-s + 0.453·11-s − 1.61·13-s + 0.989·17-s + 0.226·19-s − 1.01·23-s − 0.112·25-s − 1.00·29-s − 1.12·31-s + 0.176·35-s − 1.07·37-s + 1.18·41-s + 0.868·43-s − 0.930·47-s − 0.964·49-s + 1.66·53-s + 0.427·55-s − 1.31·59-s + 1.12·61-s − 1.51·65-s − 1.27·67-s + 0.177·71-s + 0.709·73-s + 0.0850·77-s + 1.30·79-s − 0.0175·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 263.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 170.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.00e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.00e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.77e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.87e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.22e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.80e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.07e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.99e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.36e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.73e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.11e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.89e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.47e7T + 8.07e13T^{2} \) |
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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
−9.738752423218596536721189313832, −9.356572061916763101811895166686, −7.928844851381959415201909388802, −7.10887639156779234039084784014, −5.86530534892322098999733708174, −5.16602885728760664031776249661, −3.82794240647127882938863842840, −2.43685805270079556556249601482, −1.53153808751246876463087853030, 0,
1.53153808751246876463087853030, 2.43685805270079556556249601482, 3.82794240647127882938863842840, 5.16602885728760664031776249661, 5.86530534892322098999733708174, 7.10887639156779234039084784014, 7.928844851381959415201909388802, 9.356572061916763101811895166686, 9.738752423218596536721189313832
