L(s) = 1 | + 4.85i·2-s − 7.60·4-s + 12.5·5-s + 50.4·7-s + 40.7i·8-s + 60.7i·10-s + 191. i·11-s + 0.640i·13-s + 245. i·14-s − 319.·16-s + 549.·17-s + 524.·19-s − 95.0·20-s − 929.·22-s − 543. i·23-s + ⋯ |
L(s) = 1 | + 1.21i·2-s − 0.475·4-s + 0.500·5-s + 1.03·7-s + 0.637i·8-s + 0.607i·10-s + 1.58i·11-s + 0.00378i·13-s + 1.25i·14-s − 1.24·16-s + 1.90·17-s + 1.45·19-s − 0.237·20-s − 1.91·22-s − 1.02i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.107006569\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.107006569\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (-2.56e3 - 2.35e3i)T \) |
good | 2 | \( 1 - 4.85iT - 16T^{2} \) |
| 5 | \( 1 - 12.5T + 625T^{2} \) |
| 7 | \( 1 - 50.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 191. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 0.640iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 549.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 524.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 543. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 296.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.07e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.16e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.12e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.52e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 103. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.81e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.90e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.85e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.82e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.75e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 9.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.73e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.52e4iT - 8.85e7T^{2} \) |
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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
−10.27300947933396910360646840890, −9.678727504216698417618121125514, −8.485442741833969610133219742343, −7.59051919507562890985392296281, −7.22793766658756290044657430237, −5.88657133257462375482778506548, −5.27285523148960599224905035656, −4.34498523306217374638413020341, −2.50918772200617040590506237965, −1.34828556719389393001832645286,
0.900710323041666575098444734945, 1.55342724430119648758375968912, 2.98265312723435675075945774414, 3.65462076203458523650228605678, 5.21399274924335142540654676964, 5.88914980623416365745061218027, 7.38992025260859897357069182467, 8.221455854642131371157583704471, 9.368647144355246139074753939173, 10.03395825682435781371763257685