L(s) = 1 | + 3.75i·2-s − 10.0·4-s + 7.83i·5-s − 8.18·7-s − 22.8i·8-s − 29.4·10-s − 2.87·13-s − 30.6i·14-s + 45.3·16-s − 10.9i·17-s + 26.5·19-s − 79.0i·20-s − 17.9i·23-s − 36.4·25-s − 10.7i·26-s + ⋯ |
L(s) = 1 | + 1.87i·2-s − 2.52·4-s + 1.56i·5-s − 1.16·7-s − 2.85i·8-s − 2.94·10-s − 0.221·13-s − 2.19i·14-s + 2.83·16-s − 0.643i·17-s + 1.39·19-s − 3.95i·20-s − 0.781i·23-s − 1.45·25-s − 0.414i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4276675184\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4276675184\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 3.75iT - 4T^{2} \) |
| 5 | \( 1 - 7.83iT - 25T^{2} \) |
| 7 | \( 1 + 8.18T + 49T^{2} \) |
| 13 | \( 1 + 2.87T + 169T^{2} \) |
| 17 | \( 1 + 10.9iT - 289T^{2} \) |
| 19 | \( 1 - 26.5T + 361T^{2} \) |
| 23 | \( 1 + 17.9iT - 529T^{2} \) |
| 29 | \( 1 + 37.5iT - 841T^{2} \) |
| 31 | \( 1 + 31.0T + 961T^{2} \) |
| 37 | \( 1 + 45.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.29T + 1.84e3T^{2} \) |
| 47 | \( 1 - 24.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 8.48iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 91.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 40.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 85.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 65.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.2T + 9.40e3T^{2} \) |
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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
−9.732615396211715396525457334888, −8.850105078462959564545703418183, −7.62122268596764773727640118768, −7.26491263629300483795748279366, −6.46517446718748485820318377082, −6.01303920216448705123962796827, −4.93864441555578385429430237894, −3.68388486341762136791538718614, −2.87992215516119870663951641712, −0.16799238578181771550817270851,
0.952171109156746602673598863643, 1.89747470395708688152508741377, 3.33027657513273476178778609750, 3.83741141471802228576147579336, 5.08685790391246975247793427645, 5.52327092162039710194495831920, 7.23305518368671020560920182051, 8.553684130727291645839725993134, 8.994747151335113550190421504322, 9.699342544035962546770862540463