| L(s) = 1 | + (1.64 − 1.89i)2-s + (0.841 − 0.540i)3-s + (−0.614 − 4.27i)4-s + (−0.415 − 0.909i)5-s + (0.357 − 2.48i)6-s + (−0.959 + 0.281i)7-s + (−4.89 − 3.14i)8-s + (0.415 − 0.909i)9-s + (−2.41 − 0.708i)10-s + (0.0756 + 0.0872i)11-s + (−2.82 − 3.26i)12-s + (1.24 + 0.366i)13-s + (−1.04 + 2.28i)14-s + (−0.841 − 0.540i)15-s + (−5.75 + 1.69i)16-s + (−0.556 + 3.87i)17-s + ⋯ |
| L(s) = 1 | + (1.16 − 1.34i)2-s + (0.485 − 0.312i)3-s + (−0.307 − 2.13i)4-s + (−0.185 − 0.406i)5-s + (0.146 − 1.01i)6-s + (−0.362 + 0.106i)7-s + (−1.73 − 1.11i)8-s + (0.138 − 0.303i)9-s + (−0.762 − 0.223i)10-s + (0.0227 + 0.0263i)11-s + (−0.816 − 0.941i)12-s + (0.345 + 0.101i)13-s + (−0.279 + 0.611i)14-s + (−0.217 − 0.139i)15-s + (−1.43 + 0.422i)16-s + (−0.135 + 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.644451 - 2.65339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.644451 - 2.65339i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.67 - 1.07i)T \) |
| good | 2 | \( 1 + (-1.64 + 1.89i)T + (-0.284 - 1.97i)T^{2} \) |
| 5 | \( 1 + (0.415 + 0.909i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-0.0756 - 0.0872i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.24 - 0.366i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.556 - 3.87i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.195 - 1.35i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.291 + 2.02i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.96 + 3.19i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.215 + 0.471i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.90 - 8.55i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (5.54 - 3.56i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 + (4.20 - 1.23i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.46 - 1.01i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.23 - 2.72i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-9.18 + 10.5i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (6.24 - 7.20i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.0848 + 0.590i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (9.06 + 2.66i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (5.33 - 11.6i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (4.28 - 2.75i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.21 - 7.03i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.88638362474329567175981327573, −9.932380890556536568883624531516, −9.084858239526924093975496637547, −8.043946748518616238266553753401, −6.58733794157766764080933635699, −5.61273391784012912748947645690, −4.44553460926002820326401354246, −3.60336848174369651249963354214, −2.54912812532898293153801710450, −1.26944474738682491758459565163,
2.91941146309359990949652329150, 3.70403804739330526177862885613, 4.82311772426620951149728581797, 5.65865074656576041291610233561, 7.03050216703305691811793138804, 7.15663122475639869555850805231, 8.489632817428618238233824663622, 9.194895400150449239652103994243, 10.54040174542796452726395071789, 11.52108904210203568349474427926
