L(s) = 1 | + (0.275 + 0.199i)2-s + (0.309 − 0.951i)3-s + (−0.582 − 1.79i)4-s + (3.18 − 2.31i)5-s + (0.275 − 0.199i)6-s + (−0.203 − 0.627i)7-s + (0.408 − 1.25i)8-s + (−0.809 − 0.587i)9-s + 1.34·10-s + (−0.0920 + 3.31i)11-s − 1.88·12-s + (−0.809 − 0.587i)13-s + (0.0693 − 0.213i)14-s + (−1.21 − 3.74i)15-s + (−2.68 + 1.95i)16-s + (−0.812 + 0.590i)17-s + ⋯ |
L(s) = 1 | + (0.194 + 0.141i)2-s + (0.178 − 0.549i)3-s + (−0.291 − 0.896i)4-s + (1.42 − 1.03i)5-s + (0.112 − 0.0816i)6-s + (−0.0770 − 0.237i)7-s + (0.144 − 0.444i)8-s + (−0.269 − 0.195i)9-s + 0.423·10-s + (−0.0277 + 0.999i)11-s − 0.543·12-s + (−0.224 − 0.163i)13-s + (0.0185 − 0.0570i)14-s + (−0.314 − 0.967i)15-s + (−0.671 + 0.487i)16-s + (−0.196 + 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00398 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00398 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28167 - 1.28678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28167 - 1.28678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.0920 - 3.31i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.275 - 0.199i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-3.18 + 2.31i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.203 + 0.627i)T + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (0.812 - 0.590i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.57 - 4.84i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + (0.247 + 0.762i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.03 - 2.92i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.206 - 0.637i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.545 - 1.67i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + (-1.78 + 5.50i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (9.61 + 6.98i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.106 + 0.328i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.47 - 2.52i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 + (7.84 - 5.69i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.20 - 9.85i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.15 - 3.74i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.22 + 4.52i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 + (-7.33 - 5.32i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.61853157400944816280464945354, −9.898100630413051268221270928926, −9.289818180965446049949230785341, −8.353100495930864041085139746813, −6.97907748854383952118153204506, −6.06469882721095441449052490799, −5.29278284568129284833717179312, −4.36808813584272027492435259620, −2.17912861242363216274344181522, −1.20266877200124199070807434524,
2.51514608015565367428419702312, 3.06611508656797982390788145225, 4.49541181735143746673101665449, 5.67951812028896843836189124769, 6.60744444528251415720014250270, 7.72999753348653532932728394923, 9.096426850438962995354010220889, 9.306387000092911801537720673173, 10.73905507386521386764501661349, 11.04416039999230149790193911824