| L(s) = 1 | + 57.2·3-s + 125·5-s + 225.·7-s + 1.08e3·9-s + 7.84e3·11-s + 4.17e3·13-s + 7.15e3·15-s − 2.87e4·17-s − 3.23e4·19-s + 1.28e4·21-s − 2.51e4·23-s + 1.56e4·25-s − 6.30e4·27-s + 3.51e4·29-s − 2.69e5·31-s + 4.48e5·33-s + 2.81e4·35-s + 4.58e5·37-s + 2.38e5·39-s − 5.03e5·41-s + 3.92e5·43-s + 1.35e5·45-s − 1.04e6·47-s − 7.72e5·49-s − 1.64e6·51-s + 1.60e6·53-s + 9.81e5·55-s + ⋯ |
| L(s) = 1 | + 1.22·3-s + 0.447·5-s + 0.248·7-s + 0.496·9-s + 1.77·11-s + 0.526·13-s + 0.547·15-s − 1.41·17-s − 1.08·19-s + 0.303·21-s − 0.430·23-s + 0.199·25-s − 0.616·27-s + 0.267·29-s − 1.62·31-s + 2.17·33-s + 0.110·35-s + 1.48·37-s + 0.644·39-s − 1.14·41-s + 0.753·43-s + 0.221·45-s − 1.46·47-s − 0.938·49-s − 1.73·51-s + 1.47·53-s + 0.795·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.480999058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480999058\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 3 | \( 1 - 57.2T + 2.18e3T^{2} \) |
| 7 | \( 1 - 225.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.84e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.17e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.51e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.69e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.94e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.13e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.54e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.86e6T + 8.07e13T^{2} \) |
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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
−16.75946472697210387084422220694, −15.02890938813896590976762834601, −14.23671357507304823619375357655, −13.10757540567720123780585517750, −11.27223592042214699175904580728, −9.369656403822882151329209777115, −8.492051380404573966399626272292, −6.52992300238505401606721840095, −3.94956653869545282329873810584, −1.95137028792315572973891770631,
1.95137028792315572973891770631, 3.94956653869545282329873810584, 6.52992300238505401606721840095, 8.492051380404573966399626272292, 9.369656403822882151329209777115, 11.27223592042214699175904580728, 13.10757540567720123780585517750, 14.23671357507304823619375357655, 15.02890938813896590976762834601, 16.75946472697210387084422220694
