| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 21.5i·5-s + 22.6i·8-s − 60.9·10-s − 57.9i·11-s − 304.·13-s + 64.0·16-s + 323. i·17-s + 152.·19-s + 172. i·20-s − 164·22-s + 538. i·23-s + 161.·25-s + 861. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.861i·5-s + 0.353i·8-s − 0.609·10-s − 0.479i·11-s − 1.80·13-s + 0.250·16-s + 1.11i·17-s + 0.421·19-s + 0.430i·20-s − 0.338·22-s + 1.01i·23-s + 0.257·25-s + 1.27i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.421371227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421371227\) |
| \(L(3)\) |
|
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 | \( 1 + 21.5iT - 625T^{2} \) |
| 11 | \( 1 + 57.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 304.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 323. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 152.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 538. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 83.4iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 761.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 600T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.61e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.07e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.10e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.66e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.27e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.08e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.35e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.04e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 761.T + 8.85e7T^{2} \) |
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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
−9.605646948300378736919911748764, −8.760247778867108921587704748308, −8.014204376567777347556463097038, −7.04269328428636517787634032781, −5.68215451012021899402700202690, −4.98639816092100076646509884132, −4.07192147873446100299422192472, −2.94244776727246262523199655728, −1.80801812595178563220882521624, −0.71554486659648305745554866947,
0.46086984751355748368224355813, 2.30067314370550284374548266362, 3.15867888467095128416955548410, 4.62906824658652018147000950859, 5.15019877649173145700739231750, 6.50304519281623887603441037980, 7.05572456391162396890051263908, 7.68055641337474389928199298982, 8.730033885073805056517532209327, 9.867076510124713650294360332802
