| L(s) = 1 | + 22.2·2-s + 367.·4-s + 366.·5-s − 1.55e3·7-s + 5.33e3·8-s + 8.16e3·10-s − 49.9·11-s − 1.16e3·13-s − 3.46e4·14-s + 7.16e4·16-s + 2.07e4·17-s + 5.71e4·19-s + 1.34e5·20-s − 1.11e3·22-s − 1.21e4·23-s + 5.63e4·25-s − 2.60e4·26-s − 5.72e5·28-s + 9.39e4·29-s − 1.18e4·31-s + 9.12e5·32-s + 4.62e5·34-s − 5.70e5·35-s + 4.39e5·37-s + 1.27e6·38-s + 1.95e6·40-s − 2.65e5·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.87·4-s + 1.31·5-s − 1.71·7-s + 3.68·8-s + 2.58·10-s − 0.0113·11-s − 0.147·13-s − 3.37·14-s + 4.37·16-s + 1.02·17-s + 1.91·19-s + 3.76·20-s − 0.0222·22-s − 0.208·23-s + 0.721·25-s − 0.290·26-s − 4.92·28-s + 0.715·29-s − 0.0717·31-s + 4.92·32-s + 2.01·34-s − 2.25·35-s + 1.42·37-s + 3.75·38-s + 4.83·40-s − 0.602·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.480837465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.480837465\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 22.2T + 128T^{2} \) |
| 5 | \( 1 - 366.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.55e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 49.9T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.71e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 9.39e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.65e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.88e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.44e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.02e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.79e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.56e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.87e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.01e6T + 8.07e13T^{2} \) |
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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
−11.57032360495559509539107117534, −10.08147350381457553078099940269, −9.754809402898571337582834545172, −7.48715088898912649910525656653, −6.41537625371043785116190028593, −5.89469230635575958935768532230, −4.96445357591226122466122644518, −3.35550797105331358125405805343, −2.85575295061566076455363948549, −1.41963272841160901953421992185,
1.41963272841160901953421992185, 2.85575295061566076455363948549, 3.35550797105331358125405805343, 4.96445357591226122466122644518, 5.89469230635575958935768532230, 6.41537625371043785116190028593, 7.48715088898912649910525656653, 9.754809402898571337582834545172, 10.08147350381457553078099940269, 11.57032360495559509539107117534
