| L(s) = 1 | + 2i·3-s − 62i·7-s + 239·9-s − 144·11-s + 654i·13-s − 1.19e3i·17-s − 556·19-s + 124·21-s − 2.18e3i·23-s + 964i·27-s + 1.57e3·29-s + 9.66e3·31-s − 288i·33-s − 3.53e3i·37-s − 1.30e3·39-s + ⋯ |
| L(s) = 1 | + 0.128i·3-s − 0.478i·7-s + 0.983·9-s − 0.358·11-s + 1.07i·13-s − 0.998i·17-s − 0.353·19-s + 0.0613·21-s − 0.860i·23-s + 0.254i·27-s + 0.348·29-s + 1.80·31-s − 0.0460i·33-s − 0.424i·37-s − 0.137·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.045101578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045101578\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2iT - 243T^{2} \) |
| 7 | \( 1 + 62iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 144T + 1.61e5T^{2} \) |
| 13 | \( 1 - 654iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.19e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 556T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.18e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.66e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.53e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.46e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.11e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.82e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.30e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.70e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.67e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.18e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.14e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.37e4iT - 8.58e9T^{2} \) |
| 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
−11.50530356681986053599387059638, −10.40850444598797089518759862750, −9.684002936119396943139396124786, −8.527926901348310446551325487052, −7.28921152332836508689885696657, −6.52542523305834947210084386474, −4.88004882985447229453761082669, −4.03408141555895554894454265107, −2.36430465735809208703182786767, −0.78375758120441021162171958993,
1.07627324952932537696284967181, 2.57553808458694731215732636077, 4.04358779792260971919213363906, 5.35606466802781668855102985929, 6.44749162907755445946187025842, 7.69117345006829225450581028543, 8.510733674372944701350241813802, 9.867384857351306209790747433015, 10.50727108129753746467225098981, 11.74910305589269549877272505110
