| L(s) = 1 | + (−4.28 − 13.1i)2-s + (55.2 + 40.1i)3-s + (258. − 187. i)4-s + (−237. + 731. i)5-s + (293. − 901. i)6-s + (9.12e3 − 6.62e3i)7-s + (−9.33e3 − 6.78e3i)8-s + (−4.63e3 − 1.42e4i)9-s + 1.06e4·10-s + (1.96e4 − 4.43e4i)11-s + 2.18e4·12-s + (5.91e4 + 1.81e5i)13-s + (−1.26e5 − 9.20e4i)14-s + (−4.24e4 + 3.08e4i)15-s + (1.06e3 − 3.27e3i)16-s + (−8.13e4 + 2.50e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.189 − 0.583i)2-s + (0.394 + 0.286i)3-s + (0.504 − 0.366i)4-s + (−0.169 + 0.523i)5-s + (0.0923 − 0.284i)6-s + (1.43 − 1.04i)7-s + (−0.805 − 0.585i)8-s + (−0.235 − 0.725i)9-s + 0.337·10-s + (0.405 − 0.914i)11-s + 0.303·12-s + (0.574 + 1.76i)13-s + (−0.880 − 0.640i)14-s + (−0.216 + 0.157i)15-s + (0.00406 − 0.0124i)16-s + (−0.236 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.63191 - 0.991230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63191 - 0.991230i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.96e4 + 4.43e4i)T \) |
| good | 2 | \( 1 + (4.28 + 13.1i)T + (-414. + 300. i)T^{2} \) |
| 3 | \( 1 + (-55.2 - 40.1i)T + (6.08e3 + 1.87e4i)T^{2} \) |
| 5 | \( 1 + (237. - 731. i)T + (-1.58e6 - 1.14e6i)T^{2} \) |
| 7 | \( 1 + (-9.12e3 + 6.62e3i)T + (1.24e7 - 3.83e7i)T^{2} \) |
| 13 | \( 1 + (-5.91e4 - 1.81e5i)T + (-8.57e9 + 6.23e9i)T^{2} \) |
| 17 | \( 1 + (8.13e4 - 2.50e5i)T + (-9.59e10 - 6.97e10i)T^{2} \) |
| 19 | \( 1 + (2.07e5 + 1.50e5i)T + (9.97e10 + 3.06e11i)T^{2} \) |
| 23 | \( 1 + 6.12e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + (4.93e6 - 3.58e6i)T + (4.48e12 - 1.37e13i)T^{2} \) |
| 31 | \( 1 + (-5.71e5 - 1.75e6i)T + (-2.13e13 + 1.55e13i)T^{2} \) |
| 37 | \( 1 + (6.69e6 - 4.86e6i)T + (4.01e13 - 1.23e14i)T^{2} \) |
| 41 | \( 1 + (-9.32e6 - 6.77e6i)T + (1.01e14 + 3.11e14i)T^{2} \) |
| 43 | \( 1 + 7.83e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.34e7 + 9.79e6i)T + (3.45e14 + 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-1.85e7 - 5.69e7i)T + (-2.66e15 + 1.93e15i)T^{2} \) |
| 59 | \( 1 + (-4.74e7 + 3.44e7i)T + (2.67e15 - 8.23e15i)T^{2} \) |
| 61 | \( 1 + (3.35e7 - 1.03e8i)T + (-9.46e15 - 6.87e15i)T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-4.71e7 + 1.44e8i)T + (-3.70e16 - 2.69e16i)T^{2} \) |
| 73 | \( 1 + (-1.99e8 + 1.44e8i)T + (1.81e16 - 5.59e16i)T^{2} \) |
| 79 | \( 1 + (1.70e7 + 5.25e7i)T + (-9.69e16 + 7.04e16i)T^{2} \) |
| 83 | \( 1 + (1.14e8 - 3.51e8i)T + (-1.51e17 - 1.09e17i)T^{2} \) |
| 89 | \( 1 + 4.85e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (8.82e7 + 2.71e8i)T + (-6.15e17 + 4.46e17i)T^{2} \) |
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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
−18.41107343806976404060106399436, −16.76197343415465104709103676498, −14.93467245882857342632181473993, −14.10811404111674016385462469837, −11.50419531841748031692978802840, −10.83139935344517393957448036213, −8.889828164608659634850859156454, −6.71348661936246716963574619524, −3.77750374485513190675735276312, −1.43947904924635527950262612767,
2.20593640925083301327320563416, 5.36179820584047748852291613752, 7.75544086851342790470213120937, 8.555103773863582660434269492516, 11.30157709340600550301570554251, 12.65452374610079693827783250097, 14.68103853754774227741897390635, 15.64385751877580174864258606449, 17.24985093888537684731542075227, 18.25260006188965243719597220974
