| L(s) = 1 | + (−6.97 − 5.06i)2-s + (187. − 577. i)3-s + (−609. − 1.87e3i)4-s + (−462. + 335. i)5-s + (−4.23e3 + 3.07e3i)6-s + (5.69e3 + 1.75e4i)7-s + (−1.07e4 + 3.29e4i)8-s + (−1.55e5 − 1.12e5i)9-s + 4.92e3·10-s + (−4.70e5 − 2.52e5i)11-s − 1.19e6·12-s + (−3.08e4 − 2.24e4i)13-s + (4.91e4 − 1.51e5i)14-s + (1.07e5 + 3.30e5i)15-s + (−3.02e6 + 2.20e6i)16-s + (1.70e6 − 1.23e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.154 − 0.111i)2-s + (0.446 − 1.37i)3-s + (−0.297 − 0.916i)4-s + (−0.0661 + 0.0480i)5-s + (−0.222 + 0.161i)6-s + (0.128 + 0.394i)7-s + (−0.115 + 0.355i)8-s + (−0.877 − 0.637i)9-s + 0.0155·10-s + (−0.881 − 0.473i)11-s − 1.39·12-s + (−0.0230 − 0.0167i)13-s + (0.0244 − 0.0751i)14-s + (0.0364 + 0.112i)15-s + (−0.721 + 0.524i)16-s + (0.290 − 0.211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.101479 - 1.27076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101479 - 1.27076i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.70e5 + 2.52e5i)T \) |
| good | 2 | \( 1 + (6.97 + 5.06i)T + (632. + 1.94e3i)T^{2} \) |
| 3 | \( 1 + (-187. + 577. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 5 | \( 1 + (462. - 335. i)T + (1.50e7 - 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-5.69e3 - 1.75e4i)T + (-1.59e9 + 1.16e9i)T^{2} \) |
| 13 | \( 1 + (3.08e4 + 2.24e4i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-1.70e6 + 1.23e6i)T + (1.05e13 - 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-2.97e6 + 9.15e6i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 - 2.80e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (4.22e7 + 1.30e8i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (1.58e8 + 1.15e8i)T + (7.85e15 + 2.41e16i)T^{2} \) |
| 37 | \( 1 + (4.14e7 + 1.27e8i)T + (-1.43e17 + 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-4.48e8 + 1.37e9i)T + (-4.45e17 - 3.23e17i)T^{2} \) |
| 43 | \( 1 - 1.67e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (4.24e8 - 1.30e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (3.52e9 + 2.56e9i)T + (2.86e18 + 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-7.35e8 - 2.26e9i)T + (-2.43e19 + 1.77e19i)T^{2} \) |
| 61 | \( 1 + (6.86e9 - 4.98e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 - 5.02e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (3.52e9 - 2.56e9i)T + (7.14e19 - 2.19e20i)T^{2} \) |
| 73 | \( 1 + (-4.56e9 - 1.40e10i)T + (-2.53e20 + 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-2.80e10 - 2.03e10i)T + (2.31e20 + 7.11e20i)T^{2} \) |
| 83 | \( 1 + (1.42e10 - 1.03e10i)T + (3.97e20 - 1.22e21i)T^{2} \) |
| 89 | \( 1 - 3.29e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-9.92e10 - 7.21e10i)T + (2.21e21 + 6.80e21i)T^{2} \) |
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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
−17.69605121358075206980713467172, −15.39953639142820546166375910618, −13.97805879775458716364077574301, −12.94447714070061816672823638496, −11.13515166765852469021270352728, −9.107185893341429237779481869081, −7.49296820915062259578611560392, −5.63440760376927118816474879750, −2.30917720516664581121184678963, −0.65061725497668898172610707948,
3.28693116914922353820561096912, 4.68162713185847679012433577997, 7.76218250057016893033708054337, 9.234771678753334061449468984547, 10.60421263280554516099774421162, 12.65784602726480370895310995095, 14.38363838791151718076928294914, 15.82257403134057206878031027860, 16.73936906748398012441890784595, 18.21604328716126778346977571203
