| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 42.3i·5-s + 22.6i·8-s + 119.·10-s + 183. i·11-s + 98.5·13-s + 64.0·16-s + 532. i·17-s − 296.·19-s − 339. i·20-s + 518.·22-s + 878. i·23-s − 1.17e3·25-s − 278. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.69i·5-s + 0.353i·8-s + 1.19·10-s + 1.51i·11-s + 0.583·13-s + 0.250·16-s + 1.84i·17-s − 0.822·19-s − 0.847i·20-s + 1.07·22-s + 1.65i·23-s − 1.87·25-s − 0.412i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.628647341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628647341\) |
| \(L(3)\) |
|
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 | \( 1 - 42.3iT - 625T^{2} \) |
| 11 | \( 1 - 183. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 98.5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 532. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 296.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 878. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 849. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 411.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.45e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.52e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.08e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 944. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 206. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.53e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 179.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 607.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.05e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.42e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.34e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.48e3T + 8.85e7T^{2} \) |
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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
−10.15188522516919130618714085479, −9.365291714619409677707949346955, −8.194536785659146188006186038325, −7.31090268200379399857159125345, −6.54302162219081430957589688764, −5.62036639907731074389079880316, −4.14228439186005225370740909924, −3.55280544943605348702708666243, −2.36851988808680355693828814662, −1.61724951495794460243255931726,
0.45090140712173376243437883730, 0.912438161131499013391782997081, 2.73207113284851750227467863162, 4.21995427432976118359271619212, 4.78872311383209705698667168245, 5.80026975237589017811121195153, 6.40926244400387563755552265808, 7.81376541882009048764058988299, 8.413589867941748316025539736047, 8.982800796257870645983158950355
