| L(s) = 1 | − 4.51i·2-s − 4.37·4-s − 15.3i·5-s + 19.6·7-s − 52.4i·8-s − 69.5·10-s + 179. i·11-s − 125.·13-s − 88.6i·14-s − 306.·16-s + 43.9i·17-s − 32.3·19-s + 67.4i·20-s + 811.·22-s + 791. i·23-s + ⋯ |
| L(s) = 1 | − 1.12i·2-s − 0.273·4-s − 0.615i·5-s + 0.400·7-s − 0.819i·8-s − 0.695·10-s + 1.48i·11-s − 0.739·13-s − 0.452i·14-s − 1.19·16-s + 0.151i·17-s − 0.0895·19-s + 0.168i·20-s + 1.67·22-s + 1.49i·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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9645366360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9645366360\) |
| \(L(3)\) |
|
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 | \( 1 + 4.51iT - 16T^{2} \) |
| 5 | \( 1 + 15.3iT - 625T^{2} \) |
| 7 | \( 1 - 19.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 179. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 125.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 43.9iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 32.3T + 1.30e5T^{2} \) |
| 23 | \( 1 - 791. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 430. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.59e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.36e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 578. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.70e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 207. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.56e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 6.52e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.15e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 54.5iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.11e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.73e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.27e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 173. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 773.T + 8.85e7T^{2} \) |
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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
−10.33965270176703908238712987710, −9.645201616768600673431718334497, −8.915504616638064438907083021523, −7.55934575562994300853785784099, −6.91643620890790640640890919673, −5.30864709070619270717702609546, −4.51604812957989534349259388251, −3.40247552752494576436925987993, −2.05862340995272757891656893813, −1.38174191356257967658346671635,
0.22990865184954080939205072087, 2.16706644294778006220992490872, 3.31563133008850781566135691792, 4.83356652704712037962299139494, 5.69549310459872270996703553879, 6.60893827340509031105301172652, 7.26667747394034243442491206534, 8.299766983333635813119313004901, 8.816597681723369305279274772801, 10.23370651890098467825393961134
