| L(s) = 1 | + (−101. + 73.3i)2-s + (277. + 853. i)3-s + (2.28e3 − 7.03e3i)4-s + (2.10e4 + 1.52e4i)5-s + (−9.06e4 − 6.58e4i)6-s + (1.83e5 − 5.66e5i)7-s + (−3.06e4 − 9.41e4i)8-s + (6.38e5 − 4.63e5i)9-s − 3.24e6·10-s + (−8.49e5 + 5.81e6i)11-s + 6.64e6·12-s + (2.14e7 − 1.55e7i)13-s + (2.29e7 + 7.06e7i)14-s + (−7.20e6 + 2.21e7i)15-s + (5.90e7 + 4.28e7i)16-s + (−3.88e7 − 2.82e7i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.810i)2-s + (0.219 + 0.676i)3-s + (0.279 − 0.858i)4-s + (0.601 + 0.436i)5-s + (−0.793 − 0.576i)6-s + (0.590 − 1.81i)7-s + (−0.0412 − 0.127i)8-s + (0.400 − 0.290i)9-s − 1.02·10-s + (−0.144 + 0.989i)11-s + 0.642·12-s + (1.23 − 0.895i)13-s + (0.815 + 2.50i)14-s + (−0.163 + 0.502i)15-s + (0.879 + 0.639i)16-s + (−0.390 − 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.19262 + 0.485815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19262 + 0.485815i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (8.49e5 - 5.81e6i)T \) |
| good | 2 | \( 1 + (101. - 73.3i)T + (2.53e3 - 7.79e3i)T^{2} \) |
| 3 | \( 1 + (-277. - 853. i)T + (-1.28e6 + 9.37e5i)T^{2} \) |
| 5 | \( 1 + (-2.10e4 - 1.52e4i)T + (3.77e8 + 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-1.83e5 + 5.66e5i)T + (-7.83e10 - 5.69e10i)T^{2} \) |
| 13 | \( 1 + (-2.14e7 + 1.55e7i)T + (9.35e13 - 2.88e14i)T^{2} \) |
| 17 | \( 1 + (3.88e7 + 2.82e7i)T + (3.06e15 + 9.41e15i)T^{2} \) |
| 19 | \( 1 + (-4.38e6 - 1.34e7i)T + (-3.40e16 + 2.47e16i)T^{2} \) |
| 23 | \( 1 + 4.20e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-3.80e8 + 1.17e9i)T + (-8.30e18 - 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-3.92e9 + 2.84e9i)T + (7.54e18 - 2.32e19i)T^{2} \) |
| 37 | \( 1 + (4.23e9 - 1.30e10i)T + (-1.97e20 - 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-1.02e10 - 3.16e10i)T + (-7.48e20 + 5.43e20i)T^{2} \) |
| 43 | \( 1 - 4.23e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (7.27e9 + 2.23e10i)T + (-4.41e21 + 3.20e21i)T^{2} \) |
| 53 | \( 1 + (-2.08e11 + 1.51e11i)T + (8.04e21 - 2.47e22i)T^{2} \) |
| 59 | \( 1 + (3.48e10 - 1.07e11i)T + (-8.49e22 - 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-2.27e11 - 1.65e11i)T + (5.00e22 + 1.53e23i)T^{2} \) |
| 67 | \( 1 + 6.38e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-9.28e11 - 6.74e11i)T + (3.60e23 + 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-4.39e11 + 1.35e12i)T + (-1.35e24 - 9.82e23i)T^{2} \) |
| 79 | \( 1 + (7.03e11 - 5.11e11i)T + (1.44e24 - 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-1.61e12 - 1.17e12i)T + (2.74e24 + 8.43e24i)T^{2} \) |
| 89 | \( 1 + 8.25e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.13e12 - 8.21e11i)T + (2.07e25 - 6.40e25i)T^{2} \) |
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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
−17.55058976371645575341500867042, −16.17522637664735348013113941372, −14.99357539493713311098648893652, −13.46382898776574294892400131271, −10.49960740083146984262477202349, −9.862367308671442939564279304599, −7.989457678903598418969624958020, −6.68406746661229121761541667251, −4.07271096688287228052102362213, −0.975568361510779597584082689719,
1.36306649093849395978709334053, 2.28917847918701602783721724528, 5.75700133806123416931544581223, 8.396611649384531459882370665921, 9.067065030721983258596933200031, 11.03194760149999378392910186423, 12.30459904445860000431282795246, 13.88842966954370031235557808050, 15.96332177253270364778498756045, 17.76117164327260634365456693671
