| L(s) = 1 | − 14.1i·3-s + 11.1·5-s − 49.8i·7-s − 119.·9-s + 213. i·11-s − 127.·13-s − 158. i·15-s − 299.·17-s + 286. i·19-s − 705.·21-s + 596. i·23-s + 125.·25-s + 542. i·27-s − 1.23e3·29-s − 1.04e3i·31-s + ⋯ |
| L(s) = 1 | − 1.57i·3-s + 0.447·5-s − 1.01i·7-s − 1.47·9-s + 1.76i·11-s − 0.755·13-s − 0.703i·15-s − 1.03·17-s + 0.794i·19-s − 1.59·21-s + 1.12i·23-s + 0.200·25-s + 0.744i·27-s − 1.46·29-s − 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1433882199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1433882199\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 11.1T \) |
| good | 3 | \( 1 + 14.1iT - 81T^{2} \) |
| 7 | \( 1 + 49.8iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 213. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 127.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 299.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 286. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 596. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.23e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.04e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 165.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.84e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.40e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.36e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 660.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.53e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.65e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 657. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.96e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.73e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 4.75e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.02e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.74e3T + 8.85e7T^{2} \) |
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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
−11.49446380183315422682997396816, −10.20501581423353377394604098450, −9.460673844866965704587876352550, −7.987377328709064287905747529874, −7.20783265750568158995673087260, −6.81113792698386019406358504317, −5.45740103090090143447082664528, −4.09918090628619022486447060814, −2.23124361504205485128147854767, −1.53397627785903090310511406624,
0.03990977591736968136765593375, 2.46248405619482656931365022017, 3.47365284628795689002535924566, 4.83195184362545220782284465079, 5.51210659673282033766924852386, 6.57422041498685172954624296262, 8.480259313359792291026383792279, 8.958583635827320279216868454046, 9.738834737872859493870128507006, 10.86101962013796194658361295432
