| L(s) = 1 | − 18.4i·3-s − 121. i·7-s − 98.1·9-s + 438.·11-s + 758. i·13-s − 1.53e3i·17-s + 75.8·19-s − 2.25e3·21-s + 3.69e3i·23-s − 2.67e3i·27-s − 6.32e3·29-s − 2.69e3·31-s − 8.09e3i·33-s − 7.25e3i·37-s + 1.40e4·39-s + ⋯ |
| L(s) = 1 | − 1.18i·3-s − 0.940i·7-s − 0.404·9-s + 1.09·11-s + 1.24i·13-s − 1.28i·17-s + 0.0482·19-s − 1.11·21-s + 1.45i·23-s − 0.706i·27-s − 1.39·29-s − 0.502·31-s − 1.29i·33-s − 0.870i·37-s + 1.47·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.188155818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188155818\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 18.4iT - 243T^{2} \) |
| 7 | \( 1 + 121. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 438.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 758. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.53e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 75.8T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.69e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.25e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.53e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.13e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.94e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 5.68e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.95e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.79e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.12e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.31e5iT - 8.58e9T^{2} \) |
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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
−9.227310012261502242148952978333, −7.83997872193221094124120658669, −7.12783508370269101755327335960, −6.81703682908916557367152356166, −5.67701131883187968601592215553, −4.36980618168457792043840636811, −3.53333442894507571344144792811, −1.97789232515025796814013693358, −1.31141515661413398745135259807, −0.23800286845427148393301810151,
1.39405524879414850495005144427, 2.76388351955632970202821001030, 3.76273609730116530590609580941, 4.52838320264785749885526847343, 5.61765816193101251287067761495, 6.21448315898168615620868927864, 7.55559494459828757980471147954, 8.675314713463921249668794344558, 9.076733914664570473052943469168, 10.06454879680103744162318689745
