L(s) = 1 | − i·2-s − 1.80·3-s − 4-s + 1.80i·6-s + i·8-s + 2.24·9-s − 0.445i·11-s + 1.80·12-s + 16-s + 0.445·17-s − 2.24i·18-s − 1.24i·19-s − 0.445·22-s − 1.80i·24-s − 25-s + ⋯ |
L(s) = 1 | − i·2-s − 1.80·3-s − 4-s + 1.80i·6-s + i·8-s + 2.24·9-s − 0.445i·11-s + 1.80·12-s + 16-s + 0.445·17-s − 2.24i·18-s − 1.24i·19-s − 0.445·22-s − 1.80i·24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3689022094\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3689022094\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 0.445iT - T^{2} \) |
| 17 | \( 1 - 0.445T + T^{2} \) |
| 19 | \( 1 + 1.24iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.24iT - T^{2} \) |
| 43 | \( 1 + 1.24T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.80iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.80iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.80iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.24iT - T^{2} \) |
| 89 | \( 1 + 1.80iT - T^{2} \) |
| 97 | \( 1 - 0.445iT - T^{2} \) |
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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
−9.828726926530864735369413906428, −8.914080976525595979522491109248, −7.79399582333480261234512829906, −6.76640444008377456185113842691, −5.85480136821943120430479802320, −5.16202497940046944598190370037, −4.43895161016933068441655091600, −3.34867375540695208426297111857, −1.79239556361401669613777918509, −0.43578388974520938666384226127,
1.37568731327750000742660167627, 3.74376927044344457028810021312, 4.63405789280709355670213762328, 5.39047901834141214565792634718, 6.04507291953546679472809356052, 6.67514376386714239479354363952, 7.53253344238152351824628123808, 8.253031455140236487105452209784, 9.703110604279865117863864141249, 9.956789369368755789102463803810