| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 8.15i·5-s + 2.82·8-s + 11.5i·10-s − 17.6·11-s − 1.21i·13-s + 4.00·16-s + 16.6i·17-s − 0.371i·19-s + 16.3i·20-s − 24.9·22-s − 3.02·23-s − 41.5·25-s − 1.71i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 1.63i·5-s + 0.353·8-s + 1.15i·10-s − 1.60·11-s − 0.0933i·13-s + 0.250·16-s + 0.978i·17-s − 0.0195i·19-s + 0.815i·20-s − 1.13·22-s − 0.131·23-s − 1.66·25-s − 0.0660i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.597140038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597140038\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.15iT - 25T^{2} \) |
| 11 | \( 1 + 17.6T + 121T^{2} \) |
| 13 | \( 1 + 1.21iT - 169T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 + 0.371iT - 361T^{2} \) |
| 23 | \( 1 + 3.02T + 529T^{2} \) |
| 29 | \( 1 + 2.38T + 841T^{2} \) |
| 31 | \( 1 + 1.26iT - 961T^{2} \) |
| 37 | \( 1 + 27.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.37T + 2.80e3T^{2} \) |
| 59 | \( 1 + 98.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 97.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 64.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 91.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 19.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 43.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.86iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 47.2iT - 9.40e3T^{2} \) |
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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
−10.54821624186789093228711434493, −9.874453233402290124216633185082, −8.326326714299222046260761268569, −7.57796466058180162216724612280, −6.79397975204649506128641887726, −5.98029429157726610797536738886, −5.11038919575293745135258507345, −3.78988691764880134229440141651, −2.95890654069285627084479241507, −2.11283207130781279273042856999,
0.37232157239706907574116571701, 1.85939385416775124356574256445, 3.12050547401495266717209023100, 4.45626868840280593180938025632, 5.10680879282948097075207173587, 5.64644558341960268044314668308, 7.00125942522668782181653555085, 7.963449247369418221283508418393, 8.596619304989014825099382866131, 9.605259251979623737020649443832
