| L(s) = 1 | + (−19.5 − 11.3i)2-s + (−196. − 142. i)3-s + (255. + 443. i)4-s + (3.09e3 − 1.78e3i)5-s + (2.25e3 + 5.01e3i)6-s + (1.26e4 − 2.18e4i)7-s − 1.15e4i·8-s + (1.85e4 + 5.60e4i)9-s − 8.07e4·10-s + (−1.71e5 − 9.91e4i)11-s + (1.26e4 − 1.23e5i)12-s + (1.41e5 + 2.45e5i)13-s + (−4.93e5 + 2.85e5i)14-s + (−8.63e5 − 8.82e4i)15-s + (−1.31e5 + 2.27e5i)16-s − 1.47e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.810 − 0.585i)3-s + (0.249 + 0.433i)4-s + (0.989 − 0.571i)5-s + (0.289 + 0.645i)6-s + (0.749 − 1.29i)7-s − 0.353i·8-s + (0.314 + 0.949i)9-s − 0.807·10-s + (−1.06 − 0.615i)11-s + (0.0508 − 0.497i)12-s + (0.382 + 0.662i)13-s + (−0.918 + 0.530i)14-s + (−1.13 − 0.116i)15-s + (−0.125 + 0.216i)16-s − 1.03i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0622268 - 0.855729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0622268 - 0.855729i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 + 11.3i)T \) |
| 3 | \( 1 + (196. + 142. i)T \) |
| good | 5 | \( 1 + (-3.09e3 + 1.78e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-1.26e4 + 2.18e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (1.71e5 + 9.91e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-1.41e5 - 2.45e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + 1.47e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 4.67e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-4.51e6 + 2.60e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-1.64e6 - 9.47e5i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-9.25e6 - 1.60e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 9.06e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (4.27e6 - 2.46e6i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (3.66e7 - 6.34e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (2.91e8 + 1.68e8i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 8.02e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-7.95e8 + 4.59e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-3.70e6 + 6.40e6i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-9.55e8 - 1.65e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 2.72e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.42e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-7.42e8 + 1.28e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (5.66e8 + 3.27e8i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 1.33e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-2.78e9 + 4.81e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
| 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
−16.38435884787543085059627031623, −13.80326348672491228654742529742, −12.96150198020109200605410607191, −11.20585370238956063668057998433, −10.32732223722827823179007941001, −8.385844726350783403331963201947, −6.79349976835674107933899889980, −4.92803652081556773541283554004, −1.81821332367307356582676679757, −0.50560951626531010943824420861,
2.09996638065391272464285291617, 5.24879052237759808424602116615, 6.28226063445411451552158013387, 8.497882665233609484487357319819, 10.08156948335707824794903659955, 10.95942346561221919009470910710, 12.71169475263731220418146275648, 14.92161352680884564319368457042, 15.45226074568206552695716650081, 17.27273312174851836828526504659
