| L(s) = 1 | + (9.88 − 30.4i)2-s + (205. − 149. i)3-s + (−828. − 601. i)4-s + (−2.97e3 − 9.15e3i)5-s + (−2.51e3 − 7.74e3i)6-s + (94.4 + 68.6i)7-s + (−2.65e4 + 1.92e4i)8-s + (−3.47e4 + 1.06e5i)9-s − 3.08e5·10-s + (−5.40e4 − 5.31e5i)11-s − 2.60e5·12-s + (−4.72e3 + 1.45e4i)13-s + (3.02e3 − 2.19e3i)14-s + (−1.98e6 − 1.43e6i)15-s + (3.24e5 + 9.97e5i)16-s + (1.60e5 + 4.93e5i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.489 − 0.355i)3-s + (−0.404 − 0.293i)4-s + (−0.425 − 1.31i)5-s + (−0.132 − 0.406i)6-s + (0.00212 + 0.00154i)7-s + (−0.286 + 0.207i)8-s + (−0.196 + 0.603i)9-s − 0.974·10-s + (−0.101 − 0.994i)11-s − 0.302·12-s + (−0.00352 + 0.0108i)13-s + (0.00150 − 0.00109i)14-s + (−0.673 − 0.489i)15-s + (0.0772 + 0.237i)16-s + (0.0273 + 0.0842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.291163 + 1.18662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.291163 + 1.18662i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.88 + 30.4i)T \) |
| 11 | \( 1 + (5.40e4 + 5.31e5i)T \) |
| good | 3 | \( 1 + (-205. + 149. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (2.97e3 + 9.15e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-94.4 - 68.6i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (4.72e3 - 1.45e4i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-1.60e5 - 4.93e5i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (1.09e7 - 7.97e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 + 1.75e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (6.38e7 + 4.64e7i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-6.32e6 + 1.94e7i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (5.49e8 + 3.99e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-7.53e8 + 5.47e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 9.71e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.69e9 + 1.23e9i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-9.19e8 + 2.83e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (7.54e9 + 5.48e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (1.00e8 + 3.10e8i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 4.66e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-3.08e9 - 9.49e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (-1.86e10 - 1.35e10i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-3.34e9 + 1.02e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (1.05e10 + 3.24e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 - 4.56e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (3.90e10 - 1.20e11i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
| 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
−14.24093720327549068752380570612, −13.17589306901298127986815462258, −12.22335716211317852011995175077, −10.77757230681005005669458412750, −8.919548398724033147531910324029, −8.046017258379183406172712372849, −5.49645377199733600327762568760, −3.90889494506072491503395181017, −1.99406735365243193764191436534, −0.39932614433516017494034549187,
2.81918122281743136824338118102, 4.23016081283072949835781798177, 6.40842857144961576858209091710, 7.56633706347184272581077342042, 9.208830312295453428818420374513, 10.68835931423623026707441533200, 12.32456198269342345544461665949, 14.06184100041338442655468773545, 14.99663634974356684452240115910, 15.55078223497277850751392210341
