L(s) = 1 | + (−0.938 + 2.84i)3-s + (4.88 + 1.05i)5-s − 6.81i·7-s + (−7.23 − 5.34i)9-s − 7.52i·11-s + 16.2i·13-s + (−7.58 + 12.9i)15-s − 4.11·17-s − 7.86·19-s + (19.4 + 6.39i)21-s + 19.5·23-s + (22.7 + 10.2i)25-s + (22.0 − 15.6i)27-s + 55.8i·29-s + 43.4·31-s + ⋯ |
L(s) = 1 | + (−0.312 + 0.949i)3-s + (0.977 + 0.210i)5-s − 0.973i·7-s + (−0.804 − 0.594i)9-s − 0.684i·11-s + 1.24i·13-s + (−0.505 + 0.862i)15-s − 0.242·17-s − 0.413·19-s + (0.924 + 0.304i)21-s + 0.848·23-s + (0.911 + 0.411i)25-s + (0.815 − 0.578i)27-s + 1.92i·29-s + 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.962520020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.962520020\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.938 - 2.84i)T \) |
| 5 | \( 1 + (-4.88 - 1.05i)T \) |
good | 7 | \( 1 + 6.81iT - 49T^{2} \) |
| 11 | \( 1 + 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 16.2iT - 169T^{2} \) |
| 17 | \( 1 + 4.11T + 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 - 55.8iT - 841T^{2} \) |
| 31 | \( 1 - 43.4T + 961T^{2} \) |
| 37 | \( 1 + 31.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.72T + 6.88e3T^{2} \) |
| 89 | \( 1 - 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44iT - 9.40e3T^{2} \) |
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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
−10.20882098486062699074994331955, −9.030957440375420992214082456698, −8.827673370999562407860695641352, −7.16866906944042035740630318197, −6.51537822865485482135260663297, −5.59344987371318069925118097432, −4.65488754277138717349922530051, −3.80204765665183905576434188543, −2.65459373600714464225044558581, −1.05776588815820831278498948416,
0.797755670326124786965834965045, 2.16234170622973902475085151534, 2.75280266846992939386652767783, 4.69453423012436015162207402598, 5.60177425257063500903186738479, 6.12346935150872299127890364178, 7.03801034191309636829107977998, 8.103927010835750638183620253466, 8.716113676323444785318198858676, 9.730228146244587014599872008965