L(s) = 1 | + (2.11 − 1.53i)2-s + (0.190 + 0.587i)3-s + (1.5 − 4.61i)4-s + (−0.809 − 0.587i)5-s + (1.30 + 0.951i)6-s + (0.309 − 0.951i)7-s + (−2.30 − 7.10i)8-s + (2.11 − 1.53i)9-s − 2.61·10-s + 3·12-s + (−2.61 + 1.90i)13-s + (−0.809 − 2.48i)14-s + (0.190 − 0.587i)15-s + (−7.97 − 5.79i)16-s + (6.54 + 4.75i)17-s + (2.11 − 6.51i)18-s + ⋯ |
L(s) = 1 | + (1.49 − 1.08i)2-s + (0.110 + 0.339i)3-s + (0.750 − 2.30i)4-s + (−0.361 − 0.262i)5-s + (0.534 + 0.388i)6-s + (0.116 − 0.359i)7-s + (−0.816 − 2.51i)8-s + (0.706 − 0.512i)9-s − 0.827·10-s + 0.866·12-s + (−0.726 + 0.527i)13-s + (−0.216 − 0.665i)14-s + (0.0493 − 0.151i)15-s + (−1.99 − 1.44i)16-s + (1.58 + 1.15i)17-s + (0.499 − 1.53i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71422 - 3.09356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71422 - 3.09356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-2.11 + 1.53i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.190 - 0.587i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.587i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.61 - 1.90i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.54 - 4.75i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.92 + 5.93i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + (0.736 - 2.26i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.190 - 0.138i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.763 - 2.35i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.45 - 10.6i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + (1.35 + 4.16i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.73 + 2.71i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0278 - 0.0857i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.35 - 3.16i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 + (-3.97 - 2.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.01 - 9.28i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.97 - 5.06i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.66 - 6.29i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 0.145T + 89T^{2} \) |
| 97 | \( 1 + (5.66 - 4.11i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.04659796535582496573038347007, −9.660927663782434857321570214690, −8.246919759992843201813724395019, −7.02223180922307808859179485679, −6.10916987782864699082955445520, −5.03660265613332895458054818113, −4.20794368915295616730845170086, −3.74846032577490628505919013973, −2.48057027567065174679033420795, −1.16770823161840748203840874443,
2.25857628321544497355788402971, 3.42674005381836844635576213340, 4.28719109835016709363212987371, 5.38557547408922044077493073879, 5.86936875520696242612100414061, 7.11922587761937833732602836899, 7.68239009844028756614178935271, 8.043922716179596497868736810972, 9.591285645752590241242521688063, 10.59998791950823410538612160206