| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 10.8i·5-s + 22.6i·8-s − 30.5·10-s − 230. i·11-s + 30.9·13-s + 64.0·16-s + 251. i·17-s + 583.·19-s + 86.5i·20-s − 650.·22-s − 377. i·23-s + 508.·25-s − 87.4i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.432i·5-s + 0.353i·8-s − 0.305·10-s − 1.90i·11-s + 0.182·13-s + 0.250·16-s + 0.871i·17-s + 1.61·19-s + 0.216i·20-s − 1.34·22-s − 0.712i·23-s + 0.812·25-s − 0.129i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.006142999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006142999\) |
| \(L(3)\) |
|
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 | \( 1 + 10.8iT - 625T^{2} \) |
| 11 | \( 1 + 230. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 30.9T + 2.85e4T^{2} \) |
| 17 | \( 1 - 251. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 583.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 377. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 343. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.05e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 898.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 624. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 645.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.86e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 41.7iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.88e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 251.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.65e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.58e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.40e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.42e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.07e4T + 8.85e7T^{2} \) |
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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
−9.214424633141913907020355559765, −8.494819190140716377378604662676, −7.895798649348496212077432410506, −6.43442253068179621948198835280, −5.64830527078146692665911329754, −4.68958176869050065678287998193, −3.54111732035146710650981736431, −2.83467467498851186163768288157, −1.28689769883483842379758319764, −0.54715381176496758590053891282,
1.10152032957770350248117278322, 2.50205045486187612434946063663, 3.68722673859346565565622407423, 4.85112428395428128515140969742, 5.42903809438056250638053347414, 6.91176555054132297448462423413, 7.06816245642587970274795942384, 7.997503319803835660651946046644, 9.161828201144644422355032524383, 9.764184989384600837015633449990
