| L(s) = 1 | − 16.6i·3-s − 1.78·5-s − 31.2·7-s − 194.·9-s − 10.8i·11-s − 112. i·13-s + 29.6i·15-s + 202. i·17-s + 53.3·19-s + 518. i·21-s + 617. i·23-s − 621.·25-s + 1.89e3i·27-s − 1.18e3i·29-s + (657. + 700. i)31-s + ⋯ |
| L(s) = 1 | − 1.84i·3-s − 0.0715·5-s − 0.637·7-s − 2.40·9-s − 0.0893i·11-s − 0.664i·13-s + 0.131i·15-s + 0.700i·17-s + 0.147·19-s + 1.17i·21-s + 1.16i·23-s − 0.994·25-s + 2.59i·27-s − 1.40i·29-s + (0.684 + 0.729i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.233061 + 0.538477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233061 + 0.538477i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-657. - 700. i)T \) |
| good | 3 | \( 1 + 16.6iT - 81T^{2} \) |
| 5 | \( 1 + 1.78T + 625T^{2} \) |
| 7 | \( 1 + 31.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 10.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 112. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 202. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 53.3T + 1.30e5T^{2} \) |
| 23 | \( 1 - 617. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.18e3iT - 7.07e5T^{2} \) |
| 37 | \( 1 + 836. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 781. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.01e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.39e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.31e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.51e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 860.T + 2.54e7T^{2} \) |
| 73 | \( 1 - 1.90e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.47e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.12e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.32e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.01e3T + 8.85e7T^{2} \) |
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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
−12.19616236815260595379663869767, −11.43503211744124954935249968404, −9.917535446638796205947897122078, −8.423940706253238179990656300956, −7.63472705503032981010267486047, −6.55055384539318862071651491548, −5.66688581672416179443983788055, −3.26593284226814587538555079985, −1.77153200144351860556812464767, −0.23778300624461973480459828056,
2.94742046102025515406929858895, 4.13842788472156738342984316590, 5.15912710180048000753592344124, 6.55441198061753910363841973981, 8.397256255986890703634742388837, 9.445301160323103369048180370685, 10.03877974166997570281030954026, 11.09772158027132835998243686760, 12.02521524934238597742462045330, 13.57933763137044955286029087721
