| L(s) = 1 | + (11.3 − 11.2i)2-s + (−33.8 − 81.8i)3-s + (2.12 − 255. i)4-s + (−249. + 603. i)5-s + (−1.30e3 − 547. i)6-s + (−2.82e3 + 2.82e3i)7-s + (−2.86e3 − 2.93e3i)8-s + (−906. + 906. i)9-s + (3.95e3 + 9.66e3i)10-s + (4.80e3 − 1.16e4i)11-s + (−2.10e4 + 8.50e3i)12-s + (1.29e4 + 3.13e4i)13-s + (−265. + 6.39e4i)14-s + 5.78e4·15-s + (−6.55e4 − 1.08e3i)16-s − 5.04e4i·17-s + ⋯ |
| L(s) = 1 | + (0.710 − 0.704i)2-s + (−0.418 − 1.01i)3-s + (0.00829 − 0.999i)4-s + (−0.399 + 0.964i)5-s + (−1.00 − 0.422i)6-s + (−1.17 + 1.17i)7-s + (−0.698 − 0.715i)8-s + (−0.138 + 0.138i)9-s + (0.395 + 0.966i)10-s + (0.328 − 0.792i)11-s + (−1.01 + 0.410i)12-s + (0.454 + 1.09i)13-s + (−0.00690 + 1.66i)14-s + 1.14·15-s + (−0.999 − 0.0165i)16-s − 0.603i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.000432509 + 0.000516015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000432509 + 0.000516015i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-11.3 + 11.2i)T \) |
| good | 3 | \( 1 + (33.8 + 81.8i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (249. - 603. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (2.82e3 - 2.82e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-4.80e3 + 1.16e4i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.29e4 - 3.13e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 5.04e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (2.28e5 - 9.44e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (2.05e5 + 2.05e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (4.63e5 - 1.91e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 8.09e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-1.09e5 + 2.64e5i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-5.84e4 + 5.84e4i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-2.27e6 + 5.48e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 3.09e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (3.58e6 + 1.48e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (7.53e6 + 3.11e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (1.94e7 - 8.06e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-9.11e5 - 2.20e6i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-2.68e7 + 2.68e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-6.37e6 + 6.37e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 1.70e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-4.92e7 + 2.04e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-4.53e7 - 4.53e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 2.34e7T + 7.83e15T^{2} \) |
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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
−13.88502484275887545843932500044, −12.60504335642300056348133409798, −11.93839108373466126814983433818, −10.77223847141944577763821041437, −9.076427631722178122627223103283, −6.63062612802738535790467290816, −6.14061819821607424103622291083, −3.58587432074504670094109751698, −2.10692754846661029309704670851, −0.00022548586729918635313807913,
3.82569566756140157879763868345, 4.56441159512005569698308407394, 6.22278454142799752240811377885, 7.85278016901271889586911276940, 9.524073573898568774473778922111, 10.86092466413028491921803710063, 12.67311953507393160646147744997, 13.23109395029244094285292184370, 15.10084558558576655442361010710, 15.86563723575053093063150503673
