| L(s) = 1 | + (−54.1 − 166. i)2-s + (−1.06e3 − 777. i)3-s + (−1.81e4 + 1.32e4i)4-s + (1.17e4 − 3.62e4i)5-s + (−7.15e4 + 2.20e5i)6-s + (−3.00e5 + 2.18e5i)7-s + (2.02e6 + 1.46e6i)8-s + (4.74e4 + 1.45e5i)9-s − 6.67e6·10-s + (−3.06e6 − 5.00e6i)11-s + 2.96e7·12-s + (−6.08e6 − 1.87e7i)13-s + (5.25e7 + 3.82e7i)14-s + (−4.07e7 + 2.96e7i)15-s + (7.82e7 − 2.40e8i)16-s + (−3.03e6 + 9.34e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.597 − 1.83i)2-s + (−0.847 − 0.615i)3-s + (−2.21 + 1.61i)4-s + (0.337 − 1.03i)5-s + (−0.625 + 1.92i)6-s + (−0.965 + 0.701i)7-s + (2.72 + 1.98i)8-s + (0.0297 + 0.0915i)9-s − 2.10·10-s + (−0.522 − 0.852i)11-s + 2.87·12-s + (−0.349 − 1.07i)13-s + (1.86 + 1.35i)14-s + (−0.923 + 0.671i)15-s + (1.16 − 3.59i)16-s + (−0.0305 + 0.0938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.0785481 - 0.00489077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0785481 - 0.00489077i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.06e6 + 5.00e6i)T \) |
| good | 2 | \( 1 + (54.1 + 166. i)T + (-6.62e3 + 4.81e3i)T^{2} \) |
| 3 | \( 1 + (1.06e3 + 777. i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-1.17e4 + 3.62e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (3.00e5 - 2.18e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (6.08e6 + 1.87e7i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (3.03e6 - 9.34e6i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-2.81e7 - 2.04e7i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 - 3.91e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-8.54e8 + 6.20e8i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-3.56e8 - 1.09e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-9.43e9 + 6.85e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-3.13e10 - 2.27e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 + 7.46e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.72e10 + 1.98e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (4.48e10 + 1.38e11i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-1.59e11 + 1.15e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (2.79e10 - 8.61e10i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 - 1.28e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (5.55e11 - 1.70e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (1.50e12 - 1.09e12i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-9.87e11 - 3.03e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.43e11 + 1.36e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 + 8.05e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.24e12 + 3.82e12i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−16.56334386990143805686297479549, −13.00236107010579014889788188914, −12.71061064598632191970160338449, −11.40983262349446139782852837408, −9.818839634100900086049178676910, −8.518935208158695428820542904979, −5.40938506556394385931510449489, −2.95998704363526756824531483099, −1.03818973756876646198826038084, −0.06006341387147745107211373688,
4.68567121637008340634226591956, 6.31947666346294379985813333776, 7.25038736584126984885090430174, 9.624885081112343277434660215420, 10.48762618547877152894043665314, 13.60100730257809906094661739859, 14.90092605163482739112901618784, 16.14553339633069047025221150820, 16.94501141403476520111698333366, 18.05058835407056276176283250163
