| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 32.4·5-s + 216·6-s − 798.·7-s + 512·8-s + 729·9-s + 259.·10-s − 937.·11-s + 1.72e3·12-s + 6.72e3·13-s − 6.38e3·14-s + 875.·15-s + 4.09e3·16-s − 2.92e4·17-s + 5.83e3·18-s − 9.14e3·19-s + 2.07e3·20-s − 2.15e4·21-s − 7.49e3·22-s − 9.14e3·23-s + 1.38e4·24-s − 7.70e4·25-s + 5.38e4·26-s + 1.96e4·27-s − 5.11e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.115·5-s + 0.408·6-s − 0.880·7-s + 0.353·8-s + 0.333·9-s + 0.0819·10-s − 0.212·11-s + 0.288·12-s + 0.849·13-s − 0.622·14-s + 0.0669·15-s + 0.250·16-s − 1.44·17-s + 0.235·18-s − 0.305·19-s + 0.0579·20-s − 0.508·21-s − 0.150·22-s − 0.156·23-s + 0.204·24-s − 0.986·25-s + 0.600·26-s + 0.192·27-s − 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 5 | \( 1 - 32.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 798.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 937.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.72e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.14e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.14e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.08e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.69e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.86e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.43e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.50e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.35e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.59e7T + 8.07e13T^{2} \) |
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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
−9.828353659352432509021994040435, −8.878536178741528392255963770862, −7.916196637554794127921082307340, −6.67315220036328456037522636900, −6.10722311246825459392540835042, −4.68831736188330135904286256678, −3.71773806382688627713462377404, −2.78060028654937716728903125879, −1.68616512749195628770833459415, 0,
1.68616512749195628770833459415, 2.78060028654937716728903125879, 3.71773806382688627713462377404, 4.68831736188330135904286256678, 6.10722311246825459392540835042, 6.67315220036328456037522636900, 7.916196637554794127921082307340, 8.878536178741528392255963770862, 9.828353659352432509021994040435
