| L(s) = 1 | + (−19.5 − 11.3i)2-s + (214. + 114. i)3-s + (255. + 443. i)4-s + (1.21e3 − 698. i)5-s + (−2.91e3 − 4.66e3i)6-s + (1.55e4 − 2.69e4i)7-s − 1.15e4i·8-s + (3.29e4 + 4.89e4i)9-s − 3.16e4·10-s + (2.34e5 + 1.35e5i)11-s + (4.28e3 + 1.24e5i)12-s + (2.76e5 + 4.79e5i)13-s + (−6.09e5 + 3.51e5i)14-s + (3.39e5 − 1.17e4i)15-s + (−1.31e5 + 2.27e5i)16-s − 5.34e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.882 + 0.469i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.374 − 0.599i)6-s + (0.924 − 1.60i)7-s − 0.353i·8-s + (0.558 + 0.829i)9-s − 0.316·10-s + (1.45 + 0.841i)11-s + (0.0172 + 0.499i)12-s + (0.746 + 1.29i)13-s + (−1.13 + 0.654i)14-s + (0.446 − 0.0154i)15-s + (−0.125 + 0.216i)16-s − 0.376i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.13178 - 0.165514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.13178 - 0.165514i\) |
| \(L(6)\) |
|
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 | \( 1 + (-214. - 114. i)T \) |
| 5 | \( 1 + (-1.21e3 + 698. i)T \) |
| good | 7 | \( 1 + (-1.55e4 + 2.69e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.34e5 - 1.35e5i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-2.76e5 - 4.79e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + 5.34e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 1.67e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + (7.47e5 - 4.31e5i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (2.04e7 + 1.17e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (2.24e6 + 3.89e6i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 5.34e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (5.45e7 - 3.15e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-2.64e7 + 4.58e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (3.24e7 + 1.87e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 - 7.23e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-9.77e8 + 5.64e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (6.92e8 - 1.19e9i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (5.79e8 + 1.00e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 - 9.14e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 4.31e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-2.08e9 + 3.61e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-5.62e9 - 3.24e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 4.48e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-4.77e9 + 8.26e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
| 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
−11.73954999722571221268076853105, −10.80675592621649724589970312697, −9.691687285958456493935387373012, −9.022640085027728684602596452186, −7.75297983642104442554066236634, −6.82965788585061291964531867972, −4.42129029263369596695820913776, −3.88094052423545441320528195261, −1.89221798272513013199450059872, −1.19834345142530573136617025710,
1.09369783259140470930784459315, 2.06118900199032833757076281835, 3.38839971422109551528857736005, 5.56633489831995772098759720273, 6.44028301774206772129285616975, 8.020833892863070627759994257821, 8.665272126418494135786560570119, 9.422606544861497531187904782373, 11.00841387062985709345146860969, 12.03920342863869087147983624116
