L(s) = 1 | + 4.01e3·2-s + 7.74e6·4-s − 1.05e8·5-s + 3.81e9·7-s − 2.59e9·8-s − 4.21e11·10-s − 2.52e11·11-s − 3.59e12·13-s + 1.53e13·14-s − 7.53e13·16-s − 2.34e14·17-s − 6.23e14·19-s − 8.13e14·20-s − 1.01e15·22-s + 3.58e15·23-s − 8.82e14·25-s − 1.44e16·26-s + 2.95e16·28-s + 2.05e16·29-s + 1.36e17·31-s − 2.80e17·32-s − 9.40e17·34-s − 4.00e17·35-s − 1.23e18·37-s − 2.50e18·38-s + 2.72e17·40-s − 1.40e18·41-s + ⋯ |
L(s) = 1 | + 1.38·2-s + 0.922·4-s − 0.962·5-s + 0.728·7-s − 0.106·8-s − 1.33·10-s − 0.266·11-s − 0.555·13-s + 1.01·14-s − 1.07·16-s − 1.65·17-s − 1.22·19-s − 0.888·20-s − 0.369·22-s + 0.785·23-s − 0.0740·25-s − 0.770·26-s + 0.672·28-s + 0.313·29-s + 0.963·31-s − 1.37·32-s − 2.29·34-s − 0.701·35-s − 1.14·37-s − 1.70·38-s + 0.102·40-s − 0.397·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(12)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{25}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 - 4.01e3T + 8.38e6T^{2} \) |
| 5 | \( 1 + 1.05e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 3.81e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 2.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 3.59e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.34e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 6.23e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.58e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 2.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.36e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.23e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 2.18e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 8.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.63e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.01e18T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.87e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.47e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 7.64e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.02e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.58e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.13e23T + 4.96e45T^{2} \) |
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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
−14.85840786671214873369200255648, −13.42206827315168010457399756975, −12.13508118028134002697742262804, −11.03131065402882068806585857207, −8.512632381608586760515883489243, −6.78775287522706808886968282456, −4.94991242423464308328279260265, −4.04637375498270240503524340812, −2.39776089745993745527527047428, 0,
2.39776089745993745527527047428, 4.04637375498270240503524340812, 4.94991242423464308328279260265, 6.78775287522706808886968282456, 8.512632381608586760515883489243, 11.03131065402882068806585857207, 12.13508118028134002697742262804, 13.42206827315168010457399756975, 14.85840786671214873369200255648