| L(s) = 1 | + 38.7·3-s + 496.·5-s + 343·7-s − 684.·9-s + 4.03e3·11-s + 1.44e4·13-s + 1.92e4·15-s − 2.67e4·17-s + 3.66e4·19-s + 1.32e4·21-s + 3.49e4·23-s + 1.68e5·25-s − 1.11e5·27-s − 3.58e4·29-s − 1.98e5·31-s + 1.56e5·33-s + 1.70e5·35-s + 1.14e5·37-s + 5.60e5·39-s + 2.45e5·41-s + 3.02e4·43-s − 3.39e5·45-s + 3.05e5·47-s + 1.17e5·49-s − 1.03e6·51-s + 1.21e6·53-s + 2.00e6·55-s + ⋯ |
| L(s) = 1 | + 0.828·3-s + 1.77·5-s + 0.377·7-s − 0.312·9-s + 0.914·11-s + 1.82·13-s + 1.47·15-s − 1.32·17-s + 1.22·19-s + 0.313·21-s + 0.598·23-s + 2.15·25-s − 1.08·27-s − 0.273·29-s − 1.19·31-s + 0.757·33-s + 0.671·35-s + 0.371·37-s + 1.51·39-s + 0.557·41-s + 0.0579·43-s − 0.555·45-s + 0.428·47-s + 0.142·49-s − 1.09·51-s + 1.11·53-s + 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.592687293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.592687293\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 38.7T + 2.18e3T^{2} \) |
| 5 | \( 1 - 496.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.44e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.49e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.58e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.98e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.02e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.05e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.21e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.53e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.50e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.69e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.14e7T + 8.07e13T^{2} \) |
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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
−9.593543712792588602672561923441, −9.029440840865250741881308379668, −8.541728039883839052785501903308, −7.08791479017151789692315424602, −6.09934699351929186066554802649, −5.46244380002331230838653235010, −3.99105605414733900900972953126, −2.88503624387076488691915598679, −1.86750051256510849543147887986, −1.13889030639344506915434861211,
1.13889030639344506915434861211, 1.86750051256510849543147887986, 2.88503624387076488691915598679, 3.99105605414733900900972953126, 5.46244380002331230838653235010, 6.09934699351929186066554802649, 7.08791479017151789692315424602, 8.541728039883839052785501903308, 9.029440840865250741881308379668, 9.593543712792588602672561923441
