| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 0.317i·5-s + 2.82·8-s + 0.448i·10-s + 2.82·11-s − 3.11i·13-s + 4.00·16-s − 17.9i·17-s − 18.7i·19-s + 0.634i·20-s + 4.00·22-s + 19.5·23-s + 24.8·25-s − 4.40i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.0634i·5-s + 0.353·8-s + 0.0448i·10-s + 0.257·11-s − 0.239i·13-s + 0.250·16-s − 1.05i·17-s − 0.986i·19-s + 0.0317i·20-s + 0.181·22-s + 0.848·23-s + 0.995·25-s − 0.169i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.109100452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109100452\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.317iT - 25T^{2} \) |
| 11 | \( 1 - 2.82T + 121T^{2} \) |
| 13 | \( 1 + 3.11iT - 169T^{2} \) |
| 17 | \( 1 + 17.9iT - 289T^{2} \) |
| 19 | \( 1 + 18.7iT - 361T^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 - 43.5T + 841T^{2} \) |
| 31 | \( 1 - 36.5iT - 961T^{2} \) |
| 37 | \( 1 + 13.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.59T + 1.84e3T^{2} \) |
| 47 | \( 1 + 56.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 89.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 131.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 33.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 75.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148. iT - 9.40e3T^{2} \) |
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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
−9.976173246930586274295329967575, −9.032078956181932421572796110652, −8.177442845924047255137317348551, −6.91887295989015821618023567686, −6.64867928188341954428571935806, −5.16635917511823576150251238689, −4.76984678087627624228398029246, −3.36827771458288660250953077378, −2.58754575948225112483268231327, −0.949812879229394690609727890767,
1.25457825094413872687047287731, 2.58506199453950689293508654985, 3.74692380350638999830950218328, 4.55571265224317475181355686730, 5.64227000714808811679346990504, 6.40247872378090919574245742756, 7.29440588893147028347173986203, 8.288234584303485211577763701615, 9.089961065899880476020406274443, 10.24717664652337531777642770863
