| L(s) = 1 | + (−31.2 − 96.2i)2-s + (554. + 402. i)3-s + (−1.66e3 + 1.20e3i)4-s + (−1.87e3 + 5.78e3i)5-s + (2.14e4 − 6.59e4i)6-s + (−3.49e5 + 2.53e5i)7-s + (−5.02e5 − 3.65e5i)8-s + (−3.47e5 − 1.06e6i)9-s + 6.15e5·10-s + (−3.96e6 + 4.33e6i)11-s − 1.40e6·12-s + (6.88e6 + 2.12e7i)13-s + (3.53e7 + 2.56e7i)14-s + (−3.37e6 + 2.44e6i)15-s + (−2.46e7 + 7.58e7i)16-s + (−1.40e6 + 4.33e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 − 1.06i)2-s + (0.439 + 0.319i)3-s + (−0.203 + 0.147i)4-s + (−0.0537 + 0.165i)5-s + (0.187 − 0.577i)6-s + (−1.12 + 0.815i)7-s + (−0.677 − 0.492i)8-s + (−0.217 − 0.670i)9-s + 0.194·10-s + (−0.675 + 0.737i)11-s − 0.136·12-s + (0.395 + 1.21i)13-s + (1.25 + 0.911i)14-s + (−0.0764 + 0.0555i)15-s + (−0.367 + 1.12i)16-s + (−0.0141 + 0.0435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.227602 + 0.265936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227602 + 0.265936i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.96e6 - 4.33e6i)T \) |
| good | 2 | \( 1 + (31.2 + 96.2i)T + (-6.62e3 + 4.81e3i)T^{2} \) |
| 3 | \( 1 + (-554. - 402. i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (1.87e3 - 5.78e3i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (3.49e5 - 2.53e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-6.88e6 - 2.12e7i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (1.40e6 - 4.33e6i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (2.50e7 + 1.81e7i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 + 9.31e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.80e9 - 1.31e9i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (1.93e9 + 5.94e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (1.07e10 - 7.81e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-1.96e10 - 1.43e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 - 6.08e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (6.60e10 + 4.80e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (5.52e10 + 1.69e11i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-2.76e11 + 2.00e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-3.09e10 + 9.53e10i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 + 7.99e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-4.47e11 + 1.37e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (1.51e12 - 1.09e12i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-6.76e11 - 2.08e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-8.51e11 + 2.61e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 + 2.86e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-4.91e12 - 1.51e13i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−18.21508593819125986915317401920, −16.01897575798786068674570164870, −14.90875243769236602953963930412, −12.83687413884594501704799005239, −11.64748090922919731898045764093, −9.918708208126123954475303191394, −9.057987261978678660487858656136, −6.39175165391371530565774173662, −3.54367864596206917636068546712, −2.19189286218364258608529407748,
0.15011885503373423581038598783, 3.02671941080525250087739050998, 5.83361605056076525056175193886, 7.43289466929770767473723500219, 8.509265813907442294421097427138, 10.52860957486738556222602065911, 12.86094811359520289996989834699, 14.07573581936525109656557007324, 15.86411512728179743003303436162, 16.50914527545290326159307819005
