| L(s) = 1 | + (−21.1 − 6.86i)2-s + (669. − 486. i)3-s + (−2.91e3 − 2.11e3i)4-s + (−1.64e3 − 5.07e3i)5-s + (−1.74e4 + 5.68e3i)6-s + (1.46e4 − 2.01e4i)7-s + (1.00e5 + 1.38e5i)8-s + (4.74e4 − 1.46e5i)9-s + 1.18e5i·10-s + (−1.74e6 − 3.16e5i)11-s − 2.98e6·12-s + (−6.93e6 − 2.25e6i)13-s + (−4.47e5 + 3.25e5i)14-s + (−3.57e6 − 2.59e6i)15-s + (3.38e6 + 1.04e7i)16-s + (−1.36e6 + 4.44e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.330 − 0.107i)2-s + (0.918 − 0.667i)3-s + (−0.711 − 0.517i)4-s + (−0.105 − 0.324i)5-s + (−0.374 + 0.121i)6-s + (0.124 − 0.171i)7-s + (0.383 + 0.527i)8-s + (0.0893 − 0.274i)9-s + 0.118i·10-s + (−0.983 − 0.178i)11-s − 0.998·12-s + (−1.43 − 0.467i)13-s + (−0.0594 + 0.0431i)14-s + (−0.313 − 0.227i)15-s + (0.201 + 0.621i)16-s + (−0.0566 + 0.0184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0492208 - 0.872964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0492208 - 0.872964i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.74e6 + 3.16e5i)T \) |
| good | 2 | \( 1 + (21.1 + 6.86i)T + (3.31e3 + 2.40e3i)T^{2} \) |
| 3 | \( 1 + (-669. + 486. i)T + (1.64e5 - 5.05e5i)T^{2} \) |
| 5 | \( 1 + (1.64e3 + 5.07e3i)T + (-1.97e8 + 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-1.46e4 + 2.01e4i)T + (-4.27e9 - 1.31e10i)T^{2} \) |
| 13 | \( 1 + (6.93e6 + 2.25e6i)T + (1.88e13 + 1.36e13i)T^{2} \) |
| 17 | \( 1 + (1.36e6 - 4.44e5i)T + (4.71e14 - 3.42e14i)T^{2} \) |
| 19 | \( 1 + (2.41e7 + 3.32e7i)T + (-6.83e14 + 2.10e15i)T^{2} \) |
| 23 | \( 1 - 1.46e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-1.29e7 + 1.78e7i)T + (-1.09e17 - 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-1.80e8 + 5.55e8i)T + (-6.37e17 - 4.62e17i)T^{2} \) |
| 37 | \( 1 + (1.99e8 + 1.44e8i)T + (2.03e18 + 6.26e18i)T^{2} \) |
| 41 | \( 1 + (-1.35e8 - 1.86e8i)T + (-6.97e18 + 2.14e19i)T^{2} \) |
| 43 | \( 1 + 9.67e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-9.85e9 + 7.16e9i)T + (3.59e19 - 1.10e20i)T^{2} \) |
| 53 | \( 1 + (-9.64e9 + 2.96e10i)T + (-3.97e20 - 2.88e20i)T^{2} \) |
| 59 | \( 1 + (4.24e10 + 3.08e10i)T + (5.49e20 + 1.69e21i)T^{2} \) |
| 61 | \( 1 + (5.57e10 - 1.81e10i)T + (2.14e21 - 1.56e21i)T^{2} \) |
| 67 | \( 1 - 1.89e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (2.83e10 + 8.71e10i)T + (-1.32e22 + 9.64e21i)T^{2} \) |
| 73 | \( 1 + (7.24e10 - 9.97e10i)T + (-7.07e21 - 2.17e22i)T^{2} \) |
| 79 | \( 1 + (7.23e10 + 2.34e10i)T + (4.78e22 + 3.47e22i)T^{2} \) |
| 83 | \( 1 + (-5.32e11 + 1.73e11i)T + (8.64e22 - 6.28e22i)T^{2} \) |
| 89 | \( 1 + 7.67e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (1.87e11 - 5.76e11i)T + (-5.61e23 - 4.07e23i)T^{2} \) |
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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
−17.16577449689075705183323013162, −15.07427009453260179342891436328, −13.84183129338540817059824660440, −12.77420766164607047227539139619, −10.42885725240756848703877675362, −8.827642828353836269303977389000, −7.64359131327051525772382772361, −4.97282470933203322588141963719, −2.35963423267221819934551520270, −0.40338331746611535940581487231,
2.88446256386235419950632259658, 4.56533576840061554626304877930, 7.60887126869425589265808271358, 8.963280668361718156086242891139, 10.15900763057415981081344163700, 12.49074909512053375625885157501, 14.13051228345714005520562251072, 15.22478445449920735027023059381, 16.79037701304456801044887263820, 18.22093834949817796193287391080
