| L(s) = 1 | − 2.33i·2-s + 10.5·4-s − 30.0·5-s − 41.7·7-s − 62.0i·8-s + 70.2i·10-s − 117. i·11-s − 118. i·13-s + 97.6i·14-s + 23.2·16-s − 263.·17-s + 373.·19-s − 316.·20-s − 274.·22-s − 503. i·23-s + ⋯ |
| L(s) = 1 | − 0.584i·2-s + 0.657·4-s − 1.20·5-s − 0.851·7-s − 0.969i·8-s + 0.702i·10-s − 0.970i·11-s − 0.700i·13-s + 0.498i·14-s + 0.0906·16-s − 0.913·17-s + 1.03·19-s − 0.790·20-s − 0.567·22-s − 0.951i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00986 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00986 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.07955323401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07955323401\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-34.3 - 3.48e3i)T \) |
| good | 2 | \( 1 + 2.33iT - 16T^{2} \) |
| 5 | \( 1 + 30.0T + 625T^{2} \) |
| 7 | \( 1 + 41.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 117. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 118. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 263.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 373.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 503. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 549.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 450. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.43e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.44e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.20e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 967. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.66e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 1.71e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.06e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.15e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 4.17e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.99e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.93e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 2.34e3iT - 8.85e7T^{2} \) |
| 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
−10.62570677657877787395662586811, −9.827510770107375663874153751892, −8.631689890195554838177530934311, −7.79721305126387382727188678020, −6.82069239663771014700124399201, −6.06106019249060997298661054967, −4.52929773665238910680211637450, −3.29962759263859527492706781291, −2.91327109731193202584471769114, −1.04730659179422944270979353750,
0.02221630745705846989186185961, 1.91665769960627153869428000715, 3.23334273267573930631696181094, 4.26838315394102448266956892363, 5.46132493550763765238298086020, 6.71426961307366504021616998470, 7.15748281198929235804427174888, 7.943570380177930606290155715366, 9.028110827030173829570067097126, 9.962446196479609730823617609770
