L(s) = 1 | + (3 + 8.48i)3-s + 37.9·5-s − 37.9·7-s + (−62.9 + 50.9i)9-s − 134·11-s − 321. i·13-s + (113. + 321. i)15-s − 237. i·17-s − 492. i·19-s + (−113. − 321. i)21-s − 643. i·23-s + 815·25-s + (−620. − 381. i)27-s + 796.·29-s − 1.10e3·31-s + ⋯ |
L(s) = 1 | + (0.333 + 0.942i)3-s + 1.51·5-s − 0.774·7-s + (−0.777 + 0.628i)9-s − 1.10·11-s − 1.90i·13-s + (0.505 + 1.43i)15-s − 0.822i·17-s − 1.36i·19-s + (−0.258 − 0.730i)21-s − 1.21i·23-s + 1.30·25-s + (−0.851 − 0.523i)27-s + 0.947·29-s − 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.611248559\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611248559\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3 - 8.48i)T \) |
good | 5 | \( 1 - 37.9T + 625T^{2} \) |
| 7 | \( 1 + 37.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + 134T + 1.46e4T^{2} \) |
| 13 | \( 1 + 321. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 237. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 492. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 643. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 796.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.10e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.28e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 424. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.57e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.63e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.38e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 321. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 186. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.93e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.47e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.91e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 9.06e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.85e3T + 8.85e7T^{2} \) |
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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
−10.34906007078139159262407917157, −9.763876208302444719693098171006, −8.966759957578103770553846803137, −7.927524834398723589815800719498, −6.51188032419072245110733329907, −5.45946633727135401275480781455, −4.90323681184478135873501960048, −2.97404287040297724000134835028, −2.61632683857268691447442365977, −0.39269606082347173082454543518,
1.59186520771417424785865021970, 2.22808181711146725598583871078, 3.61305712144318260605117877105, 5.45125148743136476628050100445, 6.19745583611674279194586283094, 6.94236709201376229381263197796, 8.101657487122570448260435207074, 9.217990265417971563020433551487, 9.746003325687180571942087618142, 10.78980412021008490380969416277