| L(s) = 1 | − 1.07i·2-s + 14.8·4-s − 26.1·5-s + 49.1·7-s − 33.2i·8-s + 28.1i·10-s − 190. i·11-s − 198. i·13-s − 52.9i·14-s + 201.·16-s + 81.6·17-s − 590.·19-s − 387.·20-s − 205.·22-s + 808. i·23-s + ⋯ |
| L(s) = 1 | − 0.269i·2-s + 0.927·4-s − 1.04·5-s + 1.00·7-s − 0.519i·8-s + 0.281i·10-s − 1.57i·11-s − 1.17i·13-s − 0.270i·14-s + 0.787·16-s + 0.282·17-s − 1.63·19-s − 0.969·20-s − 0.424·22-s + 1.52i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.524978052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524978052\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.01e3 + 1.73e3i)T \) |
| good | 2 | \( 1 + 1.07iT - 16T^{2} \) |
| 5 | \( 1 + 26.1T + 625T^{2} \) |
| 7 | \( 1 - 49.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 190. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 198. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 81.6T + 8.35e4T^{2} \) |
| 19 | \( 1 + 590.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 808. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 714.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.37e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.10e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.13e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 349. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.73e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.50e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 233. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.66e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.12e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 8.28e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.47e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.05e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.50e4iT - 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30795572457803309917064550383, −8.630590265769988029478620939259, −8.096604625766380590427768588286, −7.35356906757075080015004482167, −6.16097697082323911840756361808, −5.22218408313367887065977270665, −3.78423440052625469929358164603, −3.05897026404342969732838547643, −1.59413505709824299157701952619, −0.36249968851624885972137102593,
1.60409790822633123910738861318, 2.49035912785542278499592781764, 4.27430831269823916101318727119, 4.66896412307951799319319855998, 6.35997884628176490992001375565, 6.98259479860855604273796167271, 7.944247049539694175403863949812, 8.417554286945792004697527082282, 9.853047266758032324099591138815, 10.78798859715871763378015135124
