L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 319.·5-s − 216·6-s + 134.·7-s + 512·8-s + 729·9-s + 2.55e3·10-s + 2.91e3·11-s − 1.72e3·12-s − 1.03e4·13-s + 1.07e3·14-s − 8.61e3·15-s + 4.09e3·16-s − 9.12e3·17-s + 5.83e3·18-s − 3.27e4·19-s + 2.04e4·20-s − 3.62e3·21-s + 2.33e4·22-s − 8.66e4·23-s − 1.38e4·24-s + 2.37e4·25-s − 8.27e4·26-s − 1.96e4·27-s + 8.58e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 0.147·7-s + 0.353·8-s + 0.333·9-s + 0.807·10-s + 0.660·11-s − 0.288·12-s − 1.30·13-s + 0.104·14-s − 0.659·15-s + 0.250·16-s − 0.450·17-s + 0.235·18-s − 1.09·19-s + 0.570·20-s − 0.0853·21-s + 0.467·22-s − 1.48·23-s − 0.204·24-s + 0.303·25-s − 0.923·26-s − 0.192·27-s + 0.0739·28-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 | \( 1 - 319.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 134.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.03e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 9.12e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.66e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.44e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.78e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.20e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.17e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.14e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.09e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 5.92e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.15e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.68e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.38e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.68e6T + 8.07e13T^{2} \) |
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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
−9.990137701900134886713759991631, −9.106797557193850424191812860112, −7.66381946495831637990735892366, −6.54757742525374189393992402771, −5.94866837449079838804133296006, −4.95451573621347606640853235577, −4.04540514388327734146438636250, −2.40533629615654160633141190367, −1.66509416747894784454205846096, 0,
1.66509416747894784454205846096, 2.40533629615654160633141190367, 4.04540514388327734146438636250, 4.95451573621347606640853235577, 5.94866837449079838804133296006, 6.54757742525374189393992402771, 7.66381946495831637990735892366, 9.106797557193850424191812860112, 9.990137701900134886713759991631