| L(s) = 1 | + 238.·5-s + 737. i·7-s + 1.70e3i·11-s − 4.48e3i·13-s − 1.99e4i·17-s + 8.09e3·19-s + 1.10e5·23-s − 2.13e4·25-s + 1.92e4·29-s − 7.02e4i·31-s + 1.75e5i·35-s + 4.41e5i·37-s − 3.82e5i·41-s + 3.29e5·43-s + 2.24e5·47-s + ⋯ |
| L(s) = 1 | + 0.852·5-s + 0.812i·7-s + 0.385i·11-s − 0.565i·13-s − 0.985i·17-s + 0.270·19-s + 1.89·23-s − 0.273·25-s + 0.146·29-s − 0.423i·31-s + 0.692i·35-s + 1.43i·37-s − 0.866i·41-s + 0.631·43-s + 0.315·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.732525790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.732525790\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 238.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 737. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.70e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 4.48e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.99e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 8.09e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.10e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.92e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.02e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 4.41e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.82e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 3.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.27e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.48e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.93e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.94e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.48e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 3.63e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.41e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.10e7T + 8.07e13T^{2} \) |
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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
−10.56329201739305654426792873216, −9.542695972612492934269812060500, −8.994834762616716543159447504305, −7.73030847161817988789458894215, −6.64451584724261498705802593017, −5.57987888071459497196434545798, −4.85039872618430081571078059359, −3.11153275459763534534491634319, −2.20884174988100146822685153640, −0.878920378389310442433042386648,
0.800817188397270185372260282963, 1.86254076731895253964938168930, 3.25492213915897622251583783714, 4.44286296671429738771249752205, 5.63170263052011136566272849803, 6.59040388191999604749817450194, 7.54396212981749117784755426135, 8.784669578460261787404190127939, 9.589058181558839888870160441279, 10.59414633058401112703910293447
