| L(s) = 1 | + (−11.7 + 16.1i)2-s + (−37.1 + 114. i)3-s + (−43.8 − 135. i)4-s + (106. − 77.2i)5-s + (−1.40e3 − 1.94e3i)6-s + (701. − 228. i)7-s + (−2.16e3 − 702. i)8-s + (−6.38e3 − 4.63e3i)9-s + 2.62e3i·10-s + (1.45e4 + 1.28e3i)11-s + 1.70e4·12-s + (−2.64e4 + 3.63e4i)13-s + (−4.54e3 + 1.40e4i)14-s + (4.88e3 + 1.50e4i)15-s + (6.61e4 − 4.80e4i)16-s + (4.98e3 + 6.86e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.732 + 1.00i)2-s + (−0.458 + 1.41i)3-s + (−0.171 − 0.527i)4-s + (0.170 − 0.123i)5-s + (−1.08 − 1.49i)6-s + (0.292 − 0.0949i)7-s + (−0.527 − 0.171i)8-s + (−0.972 − 0.706i)9-s + 0.262i·10-s + (0.996 + 0.0879i)11-s + 0.823·12-s + (−0.925 + 1.27i)13-s + (−0.118 + 0.364i)14-s + (0.0964 + 0.296i)15-s + (1.00 − 0.732i)16-s + (0.0597 + 0.0822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.254493 - 0.614878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.254493 - 0.614878i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.45e4 - 1.28e3i)T \) |
| good | 2 | \( 1 + (11.7 - 16.1i)T + (-79.1 - 243. i)T^{2} \) |
| 3 | \( 1 + (37.1 - 114. i)T + (-5.30e3 - 3.85e3i)T^{2} \) |
| 5 | \( 1 + (-106. + 77.2i)T + (1.20e5 - 3.71e5i)T^{2} \) |
| 7 | \( 1 + (-701. + 228. i)T + (4.66e6 - 3.38e6i)T^{2} \) |
| 13 | \( 1 + (2.64e4 - 3.63e4i)T + (-2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-4.98e3 - 6.86e3i)T + (-2.15e9 + 6.63e9i)T^{2} \) |
| 19 | \( 1 + (2.25e5 + 7.33e4i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 + 7.11e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-1.01e6 + 3.29e5i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-6.78e5 - 4.92e5i)T + (2.63e11 + 8.11e11i)T^{2} \) |
| 37 | \( 1 + (-7.59e5 - 2.33e6i)T + (-2.84e12 + 2.06e12i)T^{2} \) |
| 41 | \( 1 + (1.52e6 + 4.96e5i)T + (6.45e12 + 4.69e12i)T^{2} \) |
| 43 | \( 1 + 3.81e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-9.05e4 + 2.78e5i)T + (-1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-6.24e6 - 4.53e6i)T + (1.92e13 + 5.92e13i)T^{2} \) |
| 59 | \( 1 + (-1.11e5 - 3.42e5i)T + (-1.18e14 + 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-1.67e6 - 2.30e6i)T + (-5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 - 2.61e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (1.42e7 - 1.03e7i)T + (1.99e14 - 6.14e14i)T^{2} \) |
| 73 | \( 1 + (7.08e6 - 2.30e6i)T + (6.52e14 - 4.74e14i)T^{2} \) |
| 79 | \( 1 + (1.92e7 - 2.64e7i)T + (-4.68e14 - 1.44e15i)T^{2} \) |
| 83 | \( 1 + (9.49e6 + 1.30e7i)T + (-6.95e14 + 2.14e15i)T^{2} \) |
| 89 | \( 1 - 8.88e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.81e7 - 4.22e7i)T + (2.42e15 + 7.45e15i)T^{2} \) |
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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
−19.28076231028586299380399282779, −17.19984323948076030189886753384, −16.96606382649716855694641854232, −15.59013249231978737535415498148, −14.55374669988482072269193964036, −11.75204632685695580404497274291, −9.940869724845219671682898446115, −8.806587365394536564941181278365, −6.56427857161151177594592740582, −4.50403492028144495211964384396,
0.57831751556746060821062230886, 2.16239787325487861080428906875, 6.26879442139849896748813337476, 8.242655205338179349600953108906, 10.27700361805714491974554046309, 11.81694320931177228951761016742, 12.68158036980789867740346832537, 14.60586909326369973404626964934, 17.30093549783236324673293094306, 17.94844477243486059449144527348
