| L(s) = 1 | + 8·2-s + 3.23·3-s + 64·4-s − 312.·5-s + 25.9·6-s − 766.·7-s + 512·8-s − 2.17e3·9-s − 2.49e3·10-s + 252.·11-s + 207.·12-s − 1.06e3·13-s − 6.13e3·14-s − 1.01e3·15-s + 4.09e3·16-s − 1.87e4·17-s − 1.74e4·18-s − 6.85e3·19-s − 1.99e4·20-s − 2.48e3·21-s + 2.01e3·22-s + 1.15e4·23-s + 1.65e3·24-s + 1.95e4·25-s − 8.52e3·26-s − 1.41e4·27-s − 4.90e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0692·3-s + 0.5·4-s − 1.11·5-s + 0.0489·6-s − 0.844·7-s + 0.353·8-s − 0.995·9-s − 0.790·10-s + 0.0570·11-s + 0.0346·12-s − 0.134·13-s − 0.597·14-s − 0.0774·15-s + 0.250·16-s − 0.926·17-s − 0.703·18-s − 0.229·19-s − 0.558·20-s − 0.0585·21-s + 0.0403·22-s + 0.198·23-s + 0.0244·24-s + 0.249·25-s − 0.0950·26-s − 0.138·27-s − 0.422·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 - 3.23T + 2.18e3T^{2} \) |
| 5 | \( 1 + 312.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 766.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 252.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.06e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.87e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 1.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.62e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.52e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.92e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.09e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.32e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.37e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.90e6T + 8.07e13T^{2} \) |
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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
−14.21830421122488668237391438244, −12.93745289358173204480144830356, −11.86790942981865475035807348446, −10.85855714937440245662825922801, −8.979835451662045915112157503590, −7.49298622933437872789889365905, −6.04600433220233326194543298214, −4.22970940818352579430758329214, −2.87694526840743978921992319847, 0,
2.87694526840743978921992319847, 4.22970940818352579430758329214, 6.04600433220233326194543298214, 7.49298622933437872789889365905, 8.979835451662045915112157503590, 10.85855714937440245662825922801, 11.86790942981865475035807348446, 12.93745289358173204480144830356, 14.21830421122488668237391438244
