L(s) = 1 | + (−1.76 − 15.9i)2-s + (33.0 + 33.0i)3-s + (−249. + 56.0i)4-s + (47.4 + 47.4i)5-s + (467. − 584. i)6-s − 2.24e3·7-s + (1.33e3 + 3.87e3i)8-s + 2.18e3i·9-s + (670. − 838. i)10-s + (1.78e4 − 1.78e4i)11-s + (−1.01e4 − 6.40e3i)12-s + (−1.69e4 + 1.69e4i)13-s + (3.95e3 + 3.57e4i)14-s + 3.13e3i·15-s + (5.92e4 − 2.79e4i)16-s + 1.41e5·17-s + ⋯ |
L(s) = 1 | + (−0.110 − 0.993i)2-s + (0.408 + 0.408i)3-s + (−0.975 + 0.218i)4-s + (0.0759 + 0.0759i)5-s + (0.360 − 0.450i)6-s − 0.935·7-s + (0.324 + 0.945i)8-s + 0.333i·9-s + (0.0670 − 0.0838i)10-s + (1.22 − 1.22i)11-s + (−0.487 − 0.309i)12-s + (−0.593 + 0.593i)13-s + (0.103 + 0.930i)14-s + 0.0619i·15-s + (0.904 − 0.427i)16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.70355 - 0.657659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70355 - 0.657659i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.76 + 15.9i)T \) |
| 3 | \( 1 + (-33.0 - 33.0i)T \) |
good | 5 | \( 1 + (-47.4 - 47.4i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + 2.24e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.78e4 + 1.78e4i)T - 2.14e8iT^{2} \) |
| 13 | \( 1 + (1.69e4 - 1.69e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 - 1.41e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + (-1.65e5 - 1.65e5i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 - 6.49e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-6.71e4 + 6.71e4i)T - 5.00e11iT^{2} \) |
| 31 | \( 1 + 1.30e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.03e6 - 2.03e6i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 1.65e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (-1.66e6 + 1.66e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 9.60e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + (-1.35e6 - 1.35e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 + (4.17e6 - 4.17e6i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-5.53e6 + 5.53e6i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 + (-1.60e7 - 1.60e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 - 1.51e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 9.31e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.17e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.25e7 - 1.25e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + 4.99e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.98e7T + 7.83e15T^{2} \) |
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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
−13.91484947565885362174214249354, −12.41043302843130414214552138991, −11.50294225741151687392212789939, −9.937642780802407496704058097315, −9.427384160164590697056519033828, −7.975902175504558468925238601044, −5.87738319789288394420143484880, −3.93428574313731585790196374577, −2.97739367167228887360383888039, −1.00707479077752409887639489281,
0.975862995031568941706440110489, 3.38639509463013399921609759523, 5.21454183170506524207283277949, 6.78872460484128646940329721050, 7.54998376370476150847808074836, 9.252489660409983574779324350829, 9.820126334096645110636921060057, 12.18573151847001583877545705986, 13.06430811168766004969104358485, 14.30035016286256841485670939913