| L(s) = 1 | + (−0.129 + 0.0829i)2-s + (0.989 + 0.142i)3-s + (−0.821 + 1.79i)4-s + (3.74 − 1.09i)5-s + (−0.139 + 0.0637i)6-s + (2.59 + 0.532i)7-s + (−0.0867 − 0.603i)8-s + (0.959 + 0.281i)9-s + (−0.391 + 0.452i)10-s + (0.170 − 0.265i)11-s + (−1.06 + 1.66i)12-s + (−1.14 − 0.995i)13-s + (−0.378 + 0.146i)14-s + (3.86 − 0.555i)15-s + (−2.52 − 2.91i)16-s + (−2.42 − 5.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.0912 + 0.0586i)2-s + (0.571 + 0.0821i)3-s + (−0.410 + 0.898i)4-s + (1.67 − 0.491i)5-s + (−0.0569 + 0.0260i)6-s + (0.979 + 0.201i)7-s + (−0.0306 − 0.213i)8-s + (0.319 + 0.0939i)9-s + (−0.123 + 0.142i)10-s + (0.0515 − 0.0801i)11-s + (−0.308 + 0.479i)12-s + (−0.318 − 0.276i)13-s + (−0.101 + 0.0390i)14-s + (0.996 − 0.143i)15-s + (−0.631 − 0.729i)16-s + (−0.588 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98724 + 0.466415i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98724 + 0.466415i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 - 0.142i)T \) |
| 7 | \( 1 + (-2.59 - 0.532i)T \) |
| 23 | \( 1 + (1.47 + 4.56i)T \) |
| good | 2 | \( 1 + (0.129 - 0.0829i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (-3.74 + 1.09i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-0.170 + 0.265i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.14 + 0.995i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.42 + 5.31i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (3.19 - 6.98i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.19 - 4.81i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (4.53 - 0.651i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.789 + 2.69i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.21 - 7.53i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (7.04 + 1.01i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 8.31iT - 47T^{2} \) |
| 53 | \( 1 + (0.237 - 0.205i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.835 + 0.724i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.591 + 4.11i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (7.91 + 12.3i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.38 - 0.890i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (8.12 + 3.71i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.66 - 3.17i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.94 - 1.74i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.08 - 7.52i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (18.1 - 5.34i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.89104881728738283356821581960, −9.889990908315257336406428040883, −9.140739868079430479744302214161, −8.507619271414072174125047862395, −7.69186808469887898725800970356, −6.41866010555484638898056419146, −5.19627190332221496957257866108, −4.43744184140861743561839288889, −2.83183725507875191804749471180, −1.77352088754476962293345415327,
1.65891998122955921278171525447, 2.30822043262098086747536071186, 4.27962647765360523109404291845, 5.30703846891032559212362110624, 6.21243187903147885827952463477, 7.11513460566476274156495294719, 8.584294012026350702338830548440, 9.149210285090619345571808760164, 10.09101486119459616216870450438, 10.59214731509773997054600374498
