| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 8.92i·5-s + 22.6i·8-s + 25.2·10-s − 112. i·11-s − 86.8·13-s + 64.0·16-s − 321. i·17-s − 103.·19-s − 71.4i·20-s − 318.·22-s − 120. i·23-s + 545.·25-s + 245. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.357i·5-s + 0.353i·8-s + 0.252·10-s − 0.929i·11-s − 0.514·13-s + 0.250·16-s − 1.11i·17-s − 0.287·19-s − 0.178i·20-s − 0.657·22-s − 0.227i·23-s + 0.872·25-s + 0.363i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4000442755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4000442755\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.92iT - 625T^{2} \) |
| 11 | \( 1 + 112. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 86.8T + 2.85e4T^{2} \) |
| 17 | \( 1 + 321. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 103.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 120. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.51e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 68.3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 96.3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 273.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.22e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 378. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.61e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.67e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.30e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.48e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.73e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.61e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.19e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.05e4T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.070426991205693492184495906838, −8.486618249343741647784285007807, −7.35243784378404973764043971822, −6.51209370561964720163079397151, −5.34732547071533815591891815715, −4.54048685530084049372030515509, −3.24368702889999054280373776274, −2.66416027093403557736688051319, −1.23029071943249522708771015432, −0.094838585208685068355360449002,
1.33958807073259614359375631956, 2.66153308622186822919922557704, 4.18273997464488741319882258054, 4.71436697554423319708261780329, 5.86410184790897634271077798493, 6.60365384921698091122003653326, 7.60674983212666244201152089292, 8.226141169679613279197505246840, 9.171194943725567801253662094480, 9.916561613712001146322592630043
