| L(s) = 1 | + 8·2-s − 44.7·3-s + 64·4-s − 57.9·5-s − 358.·6-s + 343·7-s + 512·8-s − 181.·9-s − 463.·10-s + 435.·11-s − 2.86e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 2.59e3·15-s + 4.09e3·16-s + 1.08e4·17-s − 1.44e3·18-s − 2.08e4·19-s − 3.70e3·20-s − 1.53e4·21-s + 3.48e3·22-s + 7.94e4·23-s − 2.29e4·24-s − 7.47e4·25-s + 1.75e4·26-s + 1.06e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.957·3-s + 0.5·4-s − 0.207·5-s − 0.677·6-s + 0.377·7-s + 0.353·8-s − 0.0828·9-s − 0.146·10-s + 0.0986·11-s − 0.478·12-s + 0.277·13-s + 0.267·14-s + 0.198·15-s + 0.250·16-s + 0.536·17-s − 0.0585·18-s − 0.697·19-s − 0.103·20-s − 0.361·21-s + 0.0697·22-s + 1.36·23-s − 0.338·24-s − 0.957·25-s + 0.196·26-s + 1.03·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 + 44.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 57.9T + 7.81e4T^{2} \) |
| 11 | \( 1 - 435.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.08e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.08e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.94e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.54e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.80e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.38e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.49e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.58e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.69e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.11e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.39e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.81e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.79e6T + 8.07e13T^{2} \) |
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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
−11.21784281500059457938487253738, −10.23366071610294467770621475481, −8.739155755247771320445796469218, −7.50000287398680899442480247996, −6.34803636755241643946963921658, −5.47622884192920447818158628073, −4.51415468745957709434749173566, −3.17845383401902535611923208223, −1.51898917667685947433481051873, 0,
1.51898917667685947433481051873, 3.17845383401902535611923208223, 4.51415468745957709434749173566, 5.47622884192920447818158628073, 6.34803636755241643946963921658, 7.50000287398680899442480247996, 8.739155755247771320445796469218, 10.23366071610294467770621475481, 11.21784281500059457938487253738
