| L(s) = 1 | + (0.698 + 0.227i)2-s + (0.0432 + 0.0432i)3-s + (−1.18 − 0.858i)4-s + (−0.422 + 0.581i)5-s + (0.0204 + 0.0400i)6-s + (−1.42 + 2.79i)7-s + (−1.49 − 2.05i)8-s − 2.99i·9-s + (−0.426 + 0.310i)10-s + (3.06 + 0.486i)11-s + (−0.0139 − 0.0883i)12-s + (−1.79 + 0.912i)13-s + (−1.62 + 1.62i)14-s + (−0.0434 + 0.00687i)15-s + (0.325 + 1.00i)16-s + (0.304 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.494 + 0.160i)2-s + (0.0249 + 0.0249i)3-s + (−0.590 − 0.429i)4-s + (−0.188 + 0.259i)5-s + (0.00833 + 0.0163i)6-s + (−0.537 + 1.05i)7-s + (−0.528 − 0.727i)8-s − 0.998i·9-s + (−0.134 + 0.0980i)10-s + (0.925 + 0.146i)11-s + (−0.00403 − 0.0254i)12-s + (−0.496 + 0.253i)13-s + (−0.434 + 0.434i)14-s + (−0.0112 + 0.00177i)15-s + (0.0813 + 0.250i)16-s + (0.0738 − 0.466i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.835041 + 0.0436022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.835041 + 0.0436022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (2.53 + 5.88i)T \) |
| good | 2 | \( 1 + (-0.698 - 0.227i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.0432 - 0.0432i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.422 - 0.581i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (1.42 - 2.79i)T + (-4.11 - 5.66i)T^{2} \) |
| 11 | \( 1 + (-3.06 - 0.486i)T + (10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (1.79 - 0.912i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.304 + 1.92i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-3.91 - 1.99i)T + (11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (-0.275 + 0.848i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (1.43 + 9.03i)T + (-27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (5.64 - 4.10i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.49 + 2.53i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (10.3 + 3.35i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-5.24 - 10.2i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.465 - 2.94i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.37 + 4.23i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 3.69i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-8.00 + 1.26i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (0.988 + 0.156i)T + (67.5 + 21.9i)T^{2} \) |
| 73 | \( 1 - 6.49iT - 73T^{2} \) |
| 79 | \( 1 + (4.43 + 4.43i)T + 79iT^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + (6.40 - 12.5i)T + (-52.3 - 72.0i)T^{2} \) |
| 97 | \( 1 + (-6.34 + 1.00i)T + (92.2 - 29.9i)T^{2} \) |
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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
−15.75557896136827528575591072269, −14.93045676364368917801659645847, −14.07543239404372128069644458617, −12.58398797726872987664783704029, −11.78075329035360643205292152430, −9.693081714650288816138822320267, −9.057669183344745945332560103058, −6.78576236485734024867186874343, −5.51888062903332874706047919324, −3.63319350333662729485157751021,
3.60681902096579331110948601170, 5.03634743241277028996081305016, 7.18910840401283579983722550119, 8.584769799172099122491399488848, 10.06338013601090994715979606888, 11.55447024735887503749312255694, 12.85197213835531834456984739462, 13.63583780197161964440837758094, 14.62033334032451171601941743585, 16.45835626911218201509334541324
