| 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
−11.52108904210203568349474427926, −10.54040174542796452726395071789, −9.194895400150449239652103994243, −8.489632817428618238233824663622, −7.15663122475639869555850805231, −7.03050216703305691811793138804, −5.65865074656576041291610233561, −4.82311772426620951149728581797, −3.70403804739330526177862885613, −2.91941146309359990949652329150,
1.26944474738682491758459565163, 2.54912812532898293153801710450, 3.60336848174369651249963354214, 4.44553460926002820326401354246, 5.61273391784012912748947645690, 6.58733794157766764080933635699, 8.043946748518616238266553753401, 9.084858239526924093975496637547, 9.932380890556536568883624531516, 10.88638362474329567175981327573
