| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 138.·5-s + 216·6-s + 133.·7-s + 512·8-s + 729·9-s + 1.10e3·10-s + 1.80e3·11-s + 1.72e3·12-s − 1.21e4·13-s + 1.06e3·14-s + 3.73e3·15-s + 4.09e3·16-s − 3.05e4·17-s + 5.83e3·18-s − 4.37e4·19-s + 8.86e3·20-s + 3.61e3·21-s + 1.44e4·22-s − 546.·23-s + 1.38e4·24-s − 5.89e4·25-s − 9.70e4·26-s + 1.96e4·27-s + 8.55e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.495·5-s + 0.408·6-s + 0.147·7-s + 0.353·8-s + 0.333·9-s + 0.350·10-s + 0.407·11-s + 0.288·12-s − 1.53·13-s + 0.104·14-s + 0.286·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s − 1.46·19-s + 0.247·20-s + 0.0850·21-s + 0.288·22-s − 0.00936·23-s + 0.204·24-s − 0.754·25-s − 1.08·26-s + 0.192·27-s + 0.0736·28-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 | \( 1 - 138.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 133.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.80e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.21e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 546.T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.30e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.97e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.37e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.18e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.66e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.22e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.02e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.40e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.01e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.17e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.18e6T + 8.07e13T^{2} \) |
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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
−9.721447854742128487536972316841, −9.009007570343270903666777880543, −7.78005494407065440035174109458, −6.88632619922348391802847734143, −5.91015131058560911253588778610, −4.69663941728112268550585631360, −3.91854500266572717078981061976, −2.41317469193076795425020164727, −1.93482467737604388397759893180, 0,
1.93482467737604388397759893180, 2.41317469193076795425020164727, 3.91854500266572717078981061976, 4.69663941728112268550585631360, 5.91015131058560911253588778610, 6.88632619922348391802847734143, 7.78005494407065440035174109458, 9.009007570343270903666777880543, 9.721447854742128487536972316841
