| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 4.24i·5-s + 2.82i·8-s − 6·10-s − 16.9i·11-s − 8·13-s + 4.00·16-s − 12.7i·17-s + 16·19-s + 8.48i·20-s − 24·22-s + 16.9i·23-s + 7.00·25-s + 11.3i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.848i·5-s + 0.353i·8-s − 0.600·10-s − 1.54i·11-s − 0.615·13-s + 0.250·16-s − 0.748i·17-s + 0.842·19-s + 0.424i·20-s − 1.09·22-s + 0.737i·23-s + 0.280·25-s + 0.435i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8682139022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8682139022\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.24iT - 25T^{2} \) |
| 11 | \( 1 + 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 8T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 176T + 9.40e3T^{2} \) |
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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
−9.406925528344981426811008520087, −8.784479815182659277731965580058, −7.999945832806824503288140739230, −6.92627384444924372589096723994, −5.49657973238433766426055376084, −5.10652681356493178402301024112, −3.77125762750569263611071986298, −2.91145975929704736032018073339, −1.42577528943012396507964750190, −0.28912113765826791572762518793,
1.85059320603374405960128428960, 3.15559971172624801595723374441, 4.34906018890944444377569569981, 5.21975034065573553757766612376, 6.30597137565107976123829190076, 7.19682686579861573137408694641, 7.50619857545755213365572744010, 8.733354996297716235976672692723, 9.615085954447740487433903839774, 10.30043516936086576043929229061
