| L(s) = 1 | − 20.6i·3-s − 25i·5-s + 93.5·7-s − 182.·9-s − 659. i·11-s + 183. i·13-s − 515.·15-s − 467.·17-s − 225. i·19-s − 1.92e3i·21-s − 1.10e3·23-s − 625·25-s − 1.24e3i·27-s − 1.73e3i·29-s + 7.27e3·31-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s − 0.447i·5-s + 0.721·7-s − 0.751·9-s − 1.64i·11-s + 0.301i·13-s − 0.591·15-s − 0.391·17-s − 0.143i·19-s − 0.954i·21-s − 0.437·23-s − 0.200·25-s − 0.329i·27-s − 0.383i·29-s + 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.643961073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643961073\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25iT \) |
| good | 3 | \( 1 + 20.6iT - 243T^{2} \) |
| 7 | \( 1 - 93.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 659. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 183. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 467.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 225. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.42e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 4.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.23e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 619.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.69e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.06e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.19e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.26e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.55e4T + 8.58e9T^{2} \) |
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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
−10.49804657963446228097010398282, −8.964449375895424474748723157535, −8.306742524470229932014503702185, −7.50265892999537329286961849192, −6.39028545611480904013043074918, −5.55559362304644580752451978724, −4.15324052023889329068449898395, −2.54998857417611203916032150702, −1.36821672993161962648371791319, −0.43638603285141010746659671203,
1.76546795500751770666797402947, 3.18099830583243793210980006241, 4.51013445575230626740086615285, 4.88534242081382501602178345161, 6.41475804734834724867298757525, 7.56921409308401802361050868275, 8.591334589477957300006163286634, 9.894160400569350318767991063191, 10.06598032643499962502249399298, 11.15006422051646612432388032475
