| L(s) = 1 | + (−21.4 − 15.5i)2-s + (76.8 − 236. i)3-s + (58.0 + 178. i)4-s + (1.48e3 − 1.08e3i)5-s + (−5.32e3 + 3.86e3i)6-s + (−720. − 2.21e3i)7-s + (−2.64e3 + 8.15e3i)8-s + (−3.40e4 − 2.47e4i)9-s − 4.87e4·10-s + (3.89e4 + 2.89e4i)11-s + 4.67e4·12-s + (1.10e5 + 8.00e4i)13-s + (−1.90e4 + 5.86e4i)14-s + (−1.41e5 − 4.35e5i)15-s + (2.61e5 − 1.89e5i)16-s + (−1.19e5 + 8.71e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.945 − 0.687i)2-s + (0.547 − 1.68i)3-s + (0.113 + 0.349i)4-s + (1.06 − 0.774i)5-s + (−1.67 + 1.21i)6-s + (−0.113 − 0.349i)7-s + (−0.228 + 0.703i)8-s + (−1.73 − 1.25i)9-s − 1.54·10-s + (0.802 + 0.596i)11-s + 0.650·12-s + (1.06 + 0.777i)13-s + (−0.132 + 0.408i)14-s + (−0.721 − 2.22i)15-s + (0.996 − 0.724i)16-s + (−0.348 + 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.00790469 - 1.27359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00790469 - 1.27359i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.89e4 - 2.89e4i)T \) |
| good | 2 | \( 1 + (21.4 + 15.5i)T + (158. + 486. i)T^{2} \) |
| 3 | \( 1 + (-76.8 + 236. i)T + (-1.59e4 - 1.15e4i)T^{2} \) |
| 5 | \( 1 + (-1.48e3 + 1.08e3i)T + (6.03e5 - 1.85e6i)T^{2} \) |
| 7 | \( 1 + (720. + 2.21e3i)T + (-3.26e7 + 2.37e7i)T^{2} \) |
| 13 | \( 1 + (-1.10e5 - 8.00e4i)T + (3.27e9 + 1.00e10i)T^{2} \) |
| 17 | \( 1 + (1.19e5 - 8.71e4i)T + (3.66e10 - 1.12e11i)T^{2} \) |
| 19 | \( 1 + (-5.70e4 + 1.75e5i)T + (-2.61e11 - 1.89e11i)T^{2} \) |
| 23 | \( 1 + 4.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + (1.69e6 + 5.22e6i)T + (-1.17e13 + 8.52e12i)T^{2} \) |
| 31 | \( 1 + (-5.93e6 - 4.31e6i)T + (8.17e12 + 2.51e13i)T^{2} \) |
| 37 | \( 1 + (8.50e5 + 2.61e6i)T + (-1.05e14 + 7.63e13i)T^{2} \) |
| 41 | \( 1 + (5.14e6 - 1.58e7i)T + (-2.64e14 - 1.92e14i)T^{2} \) |
| 43 | \( 1 + 7.58e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (5.69e6 - 1.75e7i)T + (-9.05e14 - 6.57e14i)T^{2} \) |
| 53 | \( 1 + (9.22e7 + 6.69e7i)T + (1.01e15 + 3.13e15i)T^{2} \) |
| 59 | \( 1 + (-2.17e7 - 6.70e7i)T + (-7.00e15 + 5.09e15i)T^{2} \) |
| 61 | \( 1 + (-2.91e7 + 2.11e7i)T + (3.61e15 - 1.11e16i)T^{2} \) |
| 67 | \( 1 - 1.34e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (1.12e8 - 8.20e7i)T + (1.41e16 - 4.36e16i)T^{2} \) |
| 73 | \( 1 + (-3.83e7 - 1.18e8i)T + (-4.76e16 + 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-1.34e8 - 9.77e7i)T + (3.70e16 + 1.13e17i)T^{2} \) |
| 83 | \( 1 + (1.38e8 - 1.00e8i)T + (5.77e16 - 1.77e17i)T^{2} \) |
| 89 | \( 1 - 7.96e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (3.78e8 + 2.75e8i)T + (2.34e17 + 7.23e17i)T^{2} \) |
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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
−17.87541717666295439785220184850, −17.17567835953582738120635812290, −14.13410472152067766904982400953, −13.18507077598778582112869079729, −11.72160612964449355554231051356, −9.513935772011728079778424927877, −8.422020400770552865519245517952, −6.40173492703281519791038230612, −1.96035064769246963789262415448, −1.09125956094079384148004483728,
3.36724522796038009232326390472, 6.05320620809541038440091441003, 8.586728964781111816043495336607, 9.594003488778700498746122595437, 10.69208866752317004549414991161, 13.89327395552336652444938614037, 15.20154334163286491830173771618, 16.21199693375125041397637404275, 17.40973724422377483428376924352, 18.71475064828533967751232835836
