| L(s) = 1 | + (−64.6 + 46.9i)2-s + (−352. − 1.08e3i)3-s + (−555. + 1.71e3i)4-s + (−3.70e4 − 2.69e4i)5-s + (7.37e4 + 5.36e4i)6-s + (2.43e4 − 7.50e4i)7-s + (−2.46e5 − 7.59e5i)8-s + (2.37e5 − 1.72e5i)9-s + 3.66e6·10-s + (2.03e6 + 5.51e6i)11-s + 2.05e6·12-s + (−2.36e7 + 1.71e7i)13-s + (1.94e6 + 5.99e6i)14-s + (−1.61e7 + 4.96e7i)15-s + (3.97e7 + 2.88e7i)16-s + (1.34e8 + 9.78e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.714 + 0.519i)2-s + (−0.279 − 0.859i)3-s + (−0.0678 + 0.208i)4-s + (−1.06 − 0.770i)5-s + (0.645 + 0.469i)6-s + (0.0783 − 0.241i)7-s + (−0.332 − 1.02i)8-s + (0.148 − 0.108i)9-s + 1.15·10-s + (0.346 + 0.937i)11-s + 0.198·12-s + (−1.35 + 0.986i)13-s + (0.0691 + 0.212i)14-s + (−0.365 + 1.12i)15-s + (0.592 + 0.430i)16-s + (1.35 + 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.457845 + 0.338353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.457845 + 0.338353i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.03e6 - 5.51e6i)T \) |
| good | 2 | \( 1 + (64.6 - 46.9i)T + (2.53e3 - 7.79e3i)T^{2} \) |
| 3 | \( 1 + (352. + 1.08e3i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (3.70e4 + 2.69e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-2.43e4 + 7.50e4i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (2.36e7 - 1.71e7i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (-1.34e8 - 9.78e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (1.15e7 + 3.54e7i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 - 6.62e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (8.31e8 - 2.56e9i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-2.14e9 + 1.55e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (-3.98e9 + 1.22e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-1.15e10 - 3.56e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 + 2.38e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.47e10 + 4.53e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (1.33e11 - 9.66e10i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (1.56e11 - 4.81e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (1.09e11 + 7.94e10i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 - 4.89e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-5.01e11 - 3.64e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (1.30e11 - 4.02e11i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (8.22e11 - 5.97e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-4.22e12 - 3.07e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 3.73e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (5.35e11 - 3.88e11i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−17.31221171989943289959338026148, −16.58979880340018889403566658870, −14.99326602259256065167178005328, −12.62627222425498732703988627534, −12.11350438466246296318788674606, −9.469628483608489705755785815249, −7.82788913904698164679350785865, −7.00433722918662163669417367792, −4.24491495893233642852771904166, −1.08315268866675120144661245645,
0.42277091299947027689663543832, 3.09617586058752090014445839859, 5.18646659148422781325653732365, 7.77552174422872260362609239756, 9.665523439163124879094825530244, 10.75586570210763009319546967922, 11.81625005564487028974103021293, 14.50594464303481992612350176199, 15.52664280213431199055289620033, 17.02420480519749398010311467155
