| L(s) = 1 | − 22.2·2-s + 365.·4-s + 417.·5-s + 1.38e3·7-s − 5.28e3·8-s − 9.28e3·10-s − 3.38e3·11-s + 2.86e3·13-s − 3.08e4·14-s + 7.06e4·16-s − 3.11e3·17-s + 1.67e4·19-s + 1.52e5·20-s + 7.53e4·22-s + 1.21e4·23-s + 9.65e4·25-s − 6.36e4·26-s + 5.08e5·28-s + 1.44e5·29-s − 2.50e5·31-s − 8.93e5·32-s + 6.91e4·34-s + 5.80e5·35-s + 5.47e5·37-s − 3.71e5·38-s − 2.20e6·40-s + 7.76e5·41-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.85·4-s + 1.49·5-s + 1.53·7-s − 3.65·8-s − 2.93·10-s − 0.767·11-s + 0.361·13-s − 3.00·14-s + 4.31·16-s − 0.153·17-s + 0.558·19-s + 4.27·20-s + 1.50·22-s + 0.208·23-s + 1.23·25-s − 0.710·26-s + 4.37·28-s + 1.10·29-s − 1.50·31-s − 4.82·32-s + 0.301·34-s + 2.28·35-s + 1.77·37-s − 1.09·38-s − 5.45·40-s + 1.75·41-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
| \(L(4)\) |
\(\approx\) |
\(1.510906384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510906384\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 417.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.38e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.38e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.86e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.11e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.67e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.47e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.16e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.53e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.51e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.63e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.24e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.99e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.59e6T + 8.07e13T^{2} \) |
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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
−10.86550786996156501281646887248, −9.985573426469452291563108927035, −9.194611598235058795812800418470, −8.273437257156482918775319143017, −7.49311025571266231391817849973, −6.23892282140829957144889247819, −5.27953608025561435067425182298, −2.61274074248586162961639624172, −1.78501117922064850234135313143, −0.925701135988810145722801505226,
0.925701135988810145722801505226, 1.78501117922064850234135313143, 2.61274074248586162961639624172, 5.27953608025561435067425182298, 6.23892282140829957144889247819, 7.49311025571266231391817849973, 8.273437257156482918775319143017, 9.194611598235058795812800418470, 9.985573426469452291563108927035, 10.86550786996156501281646887248
