| L(s) = 1 | + (−79.4 − 25.8i)2-s + (−641. + 466. i)3-s + (2.33e3 + 1.69e3i)4-s + (6.73e3 + 2.07e4i)5-s + (6.30e4 − 2.04e4i)6-s + (−2.48e4 + 3.41e4i)7-s + (5.96e4 + 8.21e4i)8-s + (3.01e4 − 9.27e4i)9-s − 1.82e6i·10-s + (−1.53e6 + 8.81e5i)11-s − 2.28e6·12-s + (−4.87e6 − 1.58e6i)13-s + (2.85e6 − 2.07e6i)14-s + (−1.39e7 − 1.01e7i)15-s + (−6.26e6 − 1.92e7i)16-s + (1.85e7 − 6.03e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 − 0.403i)2-s + (−0.880 + 0.639i)3-s + (0.568 + 0.413i)4-s + (0.431 + 1.32i)5-s + (1.35 − 0.438i)6-s + (−0.211 + 0.290i)7-s + (0.227 + 0.313i)8-s + (0.0567 − 0.174i)9-s − 1.82i·10-s + (−0.867 + 0.497i)11-s − 0.765·12-s + (−1.01 − 0.328i)13-s + (0.379 − 0.275i)14-s + (−1.22 − 0.891i)15-s + (−0.373 − 1.14i)16-s + (0.770 − 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0481809 - 0.111948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0481809 - 0.111948i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.53e6 - 8.81e5i)T \) |
| good | 2 | \( 1 + (79.4 + 25.8i)T + (3.31e3 + 2.40e3i)T^{2} \) |
| 3 | \( 1 + (641. - 466. i)T + (1.64e5 - 5.05e5i)T^{2} \) |
| 5 | \( 1 + (-6.73e3 - 2.07e4i)T + (-1.97e8 + 1.43e8i)T^{2} \) |
| 7 | \( 1 + (2.48e4 - 3.41e4i)T + (-4.27e9 - 1.31e10i)T^{2} \) |
| 13 | \( 1 + (4.87e6 + 1.58e6i)T + (1.88e13 + 1.36e13i)T^{2} \) |
| 17 | \( 1 + (-1.85e7 + 6.03e6i)T + (4.71e14 - 3.42e14i)T^{2} \) |
| 19 | \( 1 + (-3.34e7 - 4.59e7i)T + (-6.83e14 + 2.10e15i)T^{2} \) |
| 23 | \( 1 + 9.37e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-3.56e8 + 4.90e8i)T + (-1.09e17 - 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-1.38e7 + 4.27e7i)T + (-6.37e17 - 4.62e17i)T^{2} \) |
| 37 | \( 1 + (3.47e9 + 2.52e9i)T + (2.03e18 + 6.26e18i)T^{2} \) |
| 41 | \( 1 + (1.73e9 + 2.38e9i)T + (-6.97e18 + 2.14e19i)T^{2} \) |
| 43 | \( 1 - 9.17e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-6.98e9 + 5.07e9i)T + (3.59e19 - 1.10e20i)T^{2} \) |
| 53 | \( 1 + (-6.37e9 + 1.96e10i)T + (-3.97e20 - 2.88e20i)T^{2} \) |
| 59 | \( 1 + (-4.52e10 - 3.28e10i)T + (5.49e20 + 1.69e21i)T^{2} \) |
| 61 | \( 1 + (-7.68e10 + 2.49e10i)T + (2.14e21 - 1.56e21i)T^{2} \) |
| 67 | \( 1 + 1.16e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (3.02e10 + 9.30e10i)T + (-1.32e22 + 9.64e21i)T^{2} \) |
| 73 | \( 1 + (7.51e10 - 1.03e11i)T + (-7.07e21 - 2.17e22i)T^{2} \) |
| 79 | \( 1 + (3.24e11 + 1.05e11i)T + (4.78e22 + 3.47e22i)T^{2} \) |
| 83 | \( 1 + (2.19e11 - 7.13e10i)T + (8.64e22 - 6.28e22i)T^{2} \) |
| 89 | \( 1 - 3.86e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-2.61e11 + 8.05e11i)T + (-5.61e23 - 4.07e23i)T^{2} \) |
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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
−18.21726888213166391111595483553, −17.39826781193219993928674765343, −16.00269044121587343118001819540, −14.32897209994423492342562715780, −11.80625565020351509878551936679, −10.27263980272925151422925551782, −10.05219706900375882635128057682, −7.58528111395306951355095052391, −5.46964794153647449009986102021, −2.46577636794213842268361139489,
0.11112414909293703682553498396, 1.14916757413986422679127707370, 5.30974530025670788611332362680, 7.12522250952277432555877187814, 8.699692165208071589006453153217, 10.11305222685862347392606413433, 12.13561092993102682236489436648, 13.34836172741979901035896580499, 16.06614144145178073732563712676, 16.97469097631587141227832998373
