| L(s) = 1 | + (−0.361 − 1.11i)2-s + (12.7 + 9.27i)3-s + (24.7 − 18.0i)4-s + (1.31 − 4.03i)5-s + (5.69 − 17.5i)6-s + (−146. + 106. i)7-s + (−59.2 − 43.0i)8-s + (1.92 + 5.91i)9-s − 4.95·10-s + (−240. − 321. i)11-s + 483.·12-s + (185. + 571. i)13-s + (171. + 124. i)14-s + (54.1 − 39.3i)15-s + (276. − 851. i)16-s + (−58.8 + 180. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0638 − 0.196i)2-s + (0.819 + 0.595i)3-s + (0.774 − 0.562i)4-s + (0.0234 − 0.0721i)5-s + (0.0646 − 0.198i)6-s + (−1.13 + 0.821i)7-s + (−0.327 − 0.237i)8-s + (0.00790 + 0.0243i)9-s − 0.0156·10-s + (−0.599 − 0.800i)11-s + 0.969·12-s + (0.304 + 0.937i)13-s + (0.233 + 0.169i)14-s + (0.0621 − 0.0451i)15-s + (0.270 − 0.831i)16-s + (−0.0493 + 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0314i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.47440 - 0.0231964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47440 - 0.0231964i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (240. + 321. i)T \) |
| good | 2 | \( 1 + (0.361 + 1.11i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-12.7 - 9.27i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + (-1.31 + 4.03i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (146. - 106. i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-185. - 571. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (58.8 - 180. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-581. - 422. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 2.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-6.32e3 + 4.59e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.09e3 - 6.44e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.49e3 + 1.81e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.01e4 + 7.35e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.29e4 - 1.66e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.98e3 + 9.19e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.71e4 - 1.97e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-512. + 1.57e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 1.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (9.85e3 - 3.03e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-9.01e3 + 6.54e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.14e4 + 6.60e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.64e4 + 5.06e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 6.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.17e4 + 6.70e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−19.57568883097657025043272237919, −18.70857249342602611389667091629, −16.15342421424022925486851518471, −15.50715101696898327779206434439, −13.99976340578520138343467674357, −12.01817293579561522053646934964, −10.17044939334078292009772038564, −8.884894140069453150304628135135, −6.21206575018476095641096736017, −2.97711443310036404219298380355,
2.90737268758298779575397509667, 6.85783362351048237728974050253, 8.073529270354010329233140567910, 10.33986439041427849550916269851, 12.52909736727876635786809965472, 13.59859211481962580762947158739, 15.42528238742783540600389540835, 16.63898782577883246920194466032, 18.23563474645392885154054636777, 20.01530657199567081305211004191
