| L(s) = 1 | + (−33.4 + 24.2i)2-s + (−18.4 − 56.9i)3-s + (369. − 1.13e3i)4-s + (−314. − 228. i)5-s + (1.99e3 + 1.45e3i)6-s + (−1.49e3 + 4.58e3i)7-s + (8.71e3 + 2.68e4i)8-s + (1.30e4 − 9.46e3i)9-s + 1.60e4·10-s + (−4.28e3 + 4.83e4i)11-s − 7.14e4·12-s + (1.25e5 − 9.11e4i)13-s + (−6.16e4 − 1.89e5i)14-s + (−7.18e3 + 2.21e4i)15-s + (−4.47e5 − 3.25e5i)16-s + (4.01e5 + 2.92e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 1.07i)2-s + (−0.131 − 0.405i)3-s + (0.720 − 2.21i)4-s + (−0.224 − 0.163i)5-s + (0.629 + 0.457i)6-s + (−0.234 + 0.722i)7-s + (0.752 + 2.31i)8-s + (0.661 − 0.480i)9-s + 0.507·10-s + (−0.0882 + 0.996i)11-s − 0.995·12-s + (1.21 − 0.884i)13-s + (−0.428 − 1.31i)14-s + (−0.0366 + 0.112i)15-s + (−1.70 − 1.24i)16-s + (1.16 + 0.848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.677402 + 0.298577i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677402 + 0.298577i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.28e3 - 4.83e4i)T \) |
| good | 2 | \( 1 + (33.4 - 24.2i)T + (158. - 486. i)T^{2} \) |
| 3 | \( 1 + (18.4 + 56.9i)T + (-1.59e4 + 1.15e4i)T^{2} \) |
| 5 | \( 1 + (314. + 228. i)T + (6.03e5 + 1.85e6i)T^{2} \) |
| 7 | \( 1 + (1.49e3 - 4.58e3i)T + (-3.26e7 - 2.37e7i)T^{2} \) |
| 13 | \( 1 + (-1.25e5 + 9.11e4i)T + (3.27e9 - 1.00e10i)T^{2} \) |
| 17 | \( 1 + (-4.01e5 - 2.92e5i)T + (3.66e10 + 1.12e11i)T^{2} \) |
| 19 | \( 1 + (2.75e4 + 8.47e4i)T + (-2.61e11 + 1.89e11i)T^{2} \) |
| 23 | \( 1 - 2.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (5.20e5 - 1.60e6i)T + (-1.17e13 - 8.52e12i)T^{2} \) |
| 31 | \( 1 + (-3.93e5 + 2.85e5i)T + (8.17e12 - 2.51e13i)T^{2} \) |
| 37 | \( 1 + (1.57e6 - 4.83e6i)T + (-1.05e14 - 7.63e13i)T^{2} \) |
| 41 | \( 1 + (5.92e6 + 1.82e7i)T + (-2.64e14 + 1.92e14i)T^{2} \) |
| 43 | \( 1 - 3.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-9.45e6 - 2.90e7i)T + (-9.05e14 + 6.57e14i)T^{2} \) |
| 53 | \( 1 + (-2.01e7 + 1.46e7i)T + (1.01e15 - 3.13e15i)T^{2} \) |
| 59 | \( 1 + (3.09e7 - 9.53e7i)T + (-7.00e15 - 5.09e15i)T^{2} \) |
| 61 | \( 1 + (2.03e7 + 1.48e7i)T + (3.61e15 + 1.11e16i)T^{2} \) |
| 67 | \( 1 + 1.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (1.68e8 + 1.22e8i)T + (1.41e16 + 4.36e16i)T^{2} \) |
| 73 | \( 1 + (9.51e7 - 2.92e8i)T + (-4.76e16 - 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-3.55e7 + 2.58e7i)T + (3.70e16 - 1.13e17i)T^{2} \) |
| 83 | \( 1 + (-1.71e7 - 1.24e7i)T + (5.77e16 + 1.77e17i)T^{2} \) |
| 89 | \( 1 - 5.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.25e9 + 9.11e8i)T + (2.34e17 - 7.23e17i)T^{2} \) |
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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
−18.32761805802411237947848788500, −17.35014005668715876604170258150, −15.81371779834545959386095306958, −15.05246578631696467901814929929, −12.56678928383536428940790334869, −10.36704633533140441202559566915, −8.896467328308093505002668517941, −7.46606807956607808306513648219, −5.96623971330781195409073543890, −1.10014382158291876348706864485,
1.03837862135505518094950900699, 3.51507505386990482473482032008, 7.45734133107828902022100605205, 9.150482807125505446479124893101, 10.54554942326332817465415710706, 11.43748076055417375649367214574, 13.38701505329048113922559847408, 16.14426365034605717448495528471, 16.84821366992094716014608745749, 18.66917846714370106779063972580
