L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (0.427 + 1.31i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + (1.69 + 2.85i)11-s − 12-s + (1.30 + 0.951i)13-s + (−0.427 + 1.31i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−2 + 1.45i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.330 + 0.239i)6-s + (0.161 + 0.496i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.316·10-s + (0.509 + 0.860i)11-s − 0.288·12-s + (0.363 + 0.263i)13-s + (−0.114 + 0.351i)14-s + (0.0797 + 0.245i)15-s + (−0.202 + 0.146i)16-s + (−0.485 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31152 + 1.22335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31152 + 1.22335i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-1.69 - 2.85i)T \) |
good | 7 | \( 1 + (-0.427 - 1.31i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2 - 1.45i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 1.53i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 + (3 + 9.23i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.85 - 2.80i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 1.53i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.95 + 6.01i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + (0.118 - 0.363i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.11 + 2.99i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.82 + 11.7i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (5.47 - 3.97i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + (-11.0 + 8.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.14 + 6.60i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (3.61 + 2.62i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (10.4 - 7.60i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-2.38 - 1.73i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.90427621702599150495230950882, −11.03849552031325068031263252403, −9.821193641547063883339468762979, −9.059000763516615612636679150287, −8.036684186803731544453204665535, −6.71058518329990958362863256918, −5.85822915768378086444413794707, −4.79377737817970356363662792510, −3.91668272618697116520026353743, −2.21023128169050730130158582728,
1.26749903171290995414401825667, 2.85694669447096723450374379971, 4.10644673390485393505708395986, 5.51470695374629689945872025419, 6.34321403608521362482391903057, 7.32812808142253993421442338603, 8.545391467611668688541069298243, 9.665830364699202144792223007981, 10.84341622620299293182020799635, 11.25386980070950861578841292062