| L(s) = 1 | + 2.01i·2-s + 11.9·4-s + 10.0i·5-s − 71.1·7-s + 56.2i·8-s − 20.3·10-s − 4.11i·11-s − 100.·13-s − 143. i·14-s + 77.9·16-s − 303. i·17-s − 418.·19-s + 120. i·20-s + 8.28·22-s − 234. i·23-s + ⋯ |
| L(s) = 1 | + 0.503i·2-s + 0.746·4-s + 0.403i·5-s − 1.45·7-s + 0.879i·8-s − 0.203·10-s − 0.0340i·11-s − 0.596·13-s − 0.730i·14-s + 0.304·16-s − 1.04i·17-s − 1.15·19-s + 0.301i·20-s + 0.0171·22-s − 0.443i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.405710122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405710122\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 - 2.01iT - 16T^{2} \) |
| 5 | \( 1 - 10.0iT - 625T^{2} \) |
| 7 | \( 1 + 71.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 4.11iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 100.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 303. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 418.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 234. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 911. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 734.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.41e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 281. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 651.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 767. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.20e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 2.77e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.16e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 5.55e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.91e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.51e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.65e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.79e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.28e3T + 8.85e7T^{2} \) |
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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
−10.12842133664314655187126981753, −9.357772164269045057307284798951, −8.188484937443458329037328415914, −7.17547866949823049241485264394, −6.55740575163010934704949749159, −5.90609259688489827766954377596, −4.52135872551016568289877419466, −3.02530801442453378972170671130, −2.40630216253661116576431527552, −0.38004982590807920251408385428,
1.05239515030329159899917670890, 2.41789253354478809321471633062, 3.31788161581589664648648629110, 4.42062673596932851512030276380, 5.97597366552708765943835259777, 6.55564228957517947164722755413, 7.51874787516308033104600884044, 8.730197782569418406976269424830, 9.655288925639932270546882930880, 10.34614099557056786248117380709
