| L(s) = 1 | − 1.41i·2-s + (2.64 − 1.41i)3-s − 2.00·4-s + (−2.00 − 3.74i)6-s − 2.64·7-s + 2.82i·8-s + (5 − 7.48i)9-s + 0.412i·11-s + (−5.29 + 2.82i)12-s − 20.5·13-s + 3.74i·14-s + 4.00·16-s − 15.8i·17-s + (−10.5 − 7.07i)18-s − 16·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.881 − 0.471i)3-s − 0.500·4-s + (−0.333 − 0.623i)6-s − 0.377·7-s + 0.353i·8-s + (0.555 − 0.831i)9-s + 0.0374i·11-s + (−0.440 + 0.235i)12-s − 1.58·13-s + 0.267i·14-s + 0.250·16-s − 0.934i·17-s + (−0.587 − 0.392i)18-s − 0.842·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3551518192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3551518192\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.64 + 1.41i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 - 0.412iT - 121T^{2} \) |
| 13 | \( 1 + 20.5T + 169T^{2} \) |
| 17 | \( 1 + 15.8iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 - 36.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.8iT - 841T^{2} \) |
| 31 | \( 1 + 5.54T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 49.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 4.12iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 31.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.8T + 9.40e3T^{2} \) |
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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
−9.308694281712743382610489060150, −8.527279302363604745478057725764, −7.40698517299004299013253678547, −7.02309269626274293473007712241, −5.58956654883324141605930513718, −4.54501983204119656618655362791, −3.44753062455993237047697742474, −2.63672426163067867782711062485, −1.67046597437368885235711929010, −0.089323955795325810027217456560,
2.07686018973427293456721163400, 3.14487399302165315813810937895, 4.32252579339097288972488757454, 4.89285594895737962145209850532, 6.19634781315296470795389560608, 6.97610517770284146105974210929, 7.955758947566936424931874181084, 8.478420583537096229548867866695, 9.385796019492369137261115332405, 10.07209649926533673295260437141
