| L(s) = 1 | − 3.37·2-s − 116.·4-s + 241.·5-s − 173.·7-s + 826.·8-s − 814.·10-s − 682.·11-s + 168.·13-s + 584.·14-s + 1.21e4·16-s − 5.56e3·17-s − 1.38e4·19-s − 2.81e4·20-s + 2.30e3·22-s + 2.46e4·23-s − 1.99e4·25-s − 567.·26-s + 2.01e4·28-s + 3.36e4·29-s + 2.27e4·31-s − 1.46e5·32-s + 1.87e4·34-s − 4.17e4·35-s + 1.81e5·37-s + 4.67e4·38-s + 1.99e5·40-s − 3.35e5·41-s + ⋯ |
| L(s) = 1 | − 0.298·2-s − 0.910·4-s + 0.862·5-s − 0.190·7-s + 0.570·8-s − 0.257·10-s − 0.154·11-s + 0.0212·13-s + 0.0569·14-s + 0.740·16-s − 0.274·17-s − 0.462·19-s − 0.785·20-s + 0.0461·22-s + 0.422·23-s − 0.255·25-s − 0.00633·26-s + 0.173·28-s + 0.256·29-s + 0.136·31-s − 0.791·32-s + 0.0820·34-s − 0.164·35-s + 0.589·37-s + 0.138·38-s + 0.492·40-s − 0.759·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 + 3.37T + 128T^{2} \) |
| 5 | \( 1 - 241.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 173.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 682.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 168.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.56e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.27e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.81e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.35e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 7.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.26e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.30e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.88e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.88e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.22e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.76e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.27e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.05e6T + 8.07e13T^{2} \) |
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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
−9.709781877435118466605261760572, −8.866014653009412210739777927815, −8.067245303352862256690625261378, −6.83493546166278422643610090348, −5.77291577521282009311851016442, −4.88178812857600409731129405393, −3.79040180666967503508356582325, −2.40118614154969101758647349965, −1.18658828312449388506593558455, 0,
1.18658828312449388506593558455, 2.40118614154969101758647349965, 3.79040180666967503508356582325, 4.88178812857600409731129405393, 5.77291577521282009311851016442, 6.83493546166278422643610090348, 8.067245303352862256690625261378, 8.866014653009412210739777927815, 9.709781877435118466605261760572
