| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 32.9i·5-s + 22.6i·8-s + 93.1·10-s + 12.7i·11-s − 31.2·13-s + 64.0·16-s − 96.6i·17-s − 487.·19-s − 263. i·20-s + 36.0·22-s − 873. i·23-s − 459.·25-s + 88.5i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.31i·5-s + 0.353i·8-s + 0.931·10-s + 0.105i·11-s − 0.185·13-s + 0.250·16-s − 0.334i·17-s − 1.35·19-s − 0.658i·20-s + 0.0745·22-s − 1.65i·23-s − 0.734·25-s + 0.130i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.639915845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639915845\) |
| \(L(3)\) |
|
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 | \( 1 - 32.9iT - 625T^{2} \) |
| 11 | \( 1 - 12.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 31.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 96.6iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 487.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 873. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.11e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 938.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.48e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.07e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.87e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.11e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 344. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.91e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.51e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.02e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.62e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.58e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.86e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.11e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.11e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.11e4T + 8.85e7T^{2} \) |
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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
−9.983867189098139962363723081893, −8.646553119887704760841684610106, −7.932779505623435181529236673346, −6.72078157452173692605571234410, −6.26109561153559722390569246131, −4.80157110823153730299062642060, −3.94533481746050150500152788317, −2.75455092878020825767533189138, −2.24150833014203574121145942983, −0.55826150931330048325198852400,
0.69040268594419688512403176202, 1.82643602818946365951038497253, 3.53516884674088119883043338071, 4.56028187986268950462667905934, 5.24186085206595232113477345617, 6.12382115480359692184156949005, 7.11276765844724187053755896569, 8.128545800570272671686697293614, 8.660179159751072990140640344494, 9.397132158474546332271314911367
