| L(s) = 1 | + (−6.57 + 20.2i)2-s + (1.76e3 − 1.28e3i)3-s + (6.26e3 + 4.54e3i)4-s + (1.37e4 + 4.22e4i)5-s + (1.43e4 + 4.42e4i)6-s + (1.26e4 + 9.20e3i)7-s + (−2.74e5 + 1.99e5i)8-s + (9.84e5 − 3.02e6i)9-s − 9.46e5·10-s + (−1.57e6 + 5.66e6i)11-s + 1.69e7·12-s + (−2.99e6 + 9.21e6i)13-s + (−2.69e5 + 1.96e5i)14-s + (7.86e7 + 5.71e7i)15-s + (1.73e7 + 5.34e7i)16-s + (−5.34e7 − 1.64e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.0726 + 0.223i)2-s + (1.40 − 1.01i)3-s + (0.764 + 0.555i)4-s + (0.393 + 1.20i)5-s + (0.125 + 0.387i)6-s + (0.0407 + 0.0295i)7-s + (−0.370 + 0.268i)8-s + (0.617 − 1.90i)9-s − 0.299·10-s + (−0.267 + 0.963i)11-s + 1.63·12-s + (−0.172 + 0.529i)13-s + (−0.00957 + 0.00695i)14-s + (1.78 + 1.29i)15-s + (0.258 + 0.796i)16-s + (−0.537 − 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.15996 + 0.646204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15996 + 0.646204i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.57e6 - 5.66e6i)T \) |
| good | 2 | \( 1 + (6.57 - 20.2i)T + (-6.62e3 - 4.81e3i)T^{2} \) |
| 3 | \( 1 + (-1.76e3 + 1.28e3i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-1.37e4 - 4.22e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-1.26e4 - 9.20e3i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (2.99e6 - 9.21e6i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (5.34e7 + 1.64e8i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-2.30e8 + 1.67e8i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 - 5.56e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (4.15e9 + 3.01e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-2.41e8 + 7.43e8i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (8.90e9 + 6.47e9i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-5.36e9 + 3.89e9i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 2.30e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (4.22e10 - 3.06e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (4.09e10 - 1.26e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (1.01e11 + 7.38e10i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-1.50e11 - 4.62e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 - 6.51e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-1.62e11 - 4.99e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (1.12e12 + 8.13e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (5.72e11 - 1.76e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (1.16e12 + 3.57e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 - 7.01e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-3.88e12 + 1.19e13i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−17.73841512889715335293397074563, −15.51174559606532935502136169051, −14.43402828068020523909971242084, −13.24792366120877717418382038301, −11.58011928396692430911842186059, −9.340824877592843277248841117886, −7.39005834708563741484147855210, −6.96438428278937146447411609716, −2.95381152996231336114022972000, −2.16337451462141505393178329386,
1.56108846405473202577290577842, 3.34011821555631753070744963962, 5.36368408109825518942829383430, 8.256918369107934917721328006508, 9.446346162995933719761275132366, 10.71256816800755930502693300844, 12.99618693358002579468744645772, 14.48704582843310322555702809550, 15.64525063061206664398469730069, 16.65534803741971205662322734108
