| L(s) = 1 | + (56.7 − 41.2i)2-s + (40.8 + 125. i)3-s + (885. − 2.72e3i)4-s + (−8.46e3 − 6.14e3i)5-s + (7.49e3 + 5.44e3i)6-s + (2.31e4 − 7.11e4i)7-s + (−1.77e4 − 5.44e4i)8-s + (1.29e5 − 9.38e4i)9-s − 7.33e5·10-s + (9.46e4 + 5.25e5i)11-s + 3.78e5·12-s + (−4.16e5 + 3.02e5i)13-s + (−1.62e6 − 4.98e6i)14-s + (4.27e5 − 1.31e6i)15-s + (1.49e6 + 1.08e6i)16-s + (−1.46e4 − 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (1.25 − 0.910i)2-s + (0.0970 + 0.298i)3-s + (0.432 − 1.33i)4-s + (−1.21 − 0.880i)5-s + (0.393 + 0.285i)6-s + (0.519 − 1.59i)7-s + (−0.191 − 0.588i)8-s + (0.729 − 0.529i)9-s − 2.31·10-s + (0.177 + 0.984i)11-s + 0.439·12-s + (−0.310 + 0.225i)13-s + (−0.805 − 2.47i)14-s + (0.145 − 0.447i)15-s + (0.357 + 0.259i)16-s + (−0.00250 − 0.00181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.51318 - 2.47683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51318 - 2.47683i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-9.46e4 - 5.25e5i)T \) |
| good | 2 | \( 1 + (-56.7 + 41.2i)T + (632. - 1.94e3i)T^{2} \) |
| 3 | \( 1 + (-40.8 - 125. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (8.46e3 + 6.14e3i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-2.31e4 + 7.11e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (4.16e5 - 3.02e5i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (1.46e4 + 1.06e4i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-3.44e6 - 1.05e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 - 1.89e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-6.15e7 + 1.89e8i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-3.13e7 + 2.27e7i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (-2.05e7 + 6.33e7i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (2.16e8 + 6.66e8i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 - 4.07e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.14e8 - 1.58e9i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (3.14e9 - 2.28e9i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (2.81e9 - 8.67e9i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-6.10e9 - 4.43e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 - 4.62e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-2.01e9 - 1.46e9i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (3.58e9 - 1.10e10i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (3.03e9 - 2.20e9i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (3.11e10 + 2.26e10i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 + 7.91e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-5.16e10 + 3.74e10i)T + (2.21e21 - 6.80e21i)T^{2} \) |
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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
−17.11644642704044324289119060534, −15.51938600446787430719608086098, −14.18000567756954378256458132700, −12.70163127875142540083874843192, −11.73230424321416707486479468392, −10.16179777065355414496441906405, −7.59707633440422882730880773392, −4.41886584542907131573704005886, −4.05680292472247052760639168345, −1.14483273748320967472828271702,
3.08278016532750743662391273886, 5.02990147848521826934651527167, 6.80073253451723518499971715541, 8.170379481146612484483767192010, 11.36036209533381116892055308533, 12.60101313397816962624451459333, 14.25115127860449852059719795575, 15.31496115927455219927039708636, 16.00500623388899947450636001912, 18.38801810175729279326850616600
