| 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 | \( 1 + (3.90e10 + 1.20e11i)T + (-5.78e21 + 4.20e21i)T^{2} \) |
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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
−15.55078223497277850751392210341, −14.99663634974356684452240115910, −14.06184100041338442655468773545, −12.32456198269342345544461665949, −10.68835931423623026707441533200, −9.208830312295453428818420374513, −7.56633706347184272581077342042, −6.40842857144961576858209091710, −4.23016081283072949835781798177, −2.81918122281743136824338118102,
0.39932614433516017494034549187, 1.99406735365243193764191436534, 3.90889494506072491503395181017, 5.49645377199733600327762568760, 8.046017258379183406172712372849, 8.919548398724033147531910324029, 10.77757230681005005669458412750, 12.22335716211317852011995175077, 13.17589306901298127986815462258, 14.24093720327549068752380570612
