L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1 − 2.44i)7-s − 0.999·8-s + 4.24·11-s + (1.12 − 1.94i)13-s + (−1.62 − 2.09i)14-s + (−0.5 + 0.866i)16-s + (1.12 + 1.94i)19-s + (2.12 − 3.67i)22-s + 1.24·23-s − 5·25-s + (−1.12 − 1.94i)26-s + (−2.62 + 0.358i)28-s + (2.12 + 3.67i)29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.377 − 0.925i)7-s − 0.353·8-s + 1.27·11-s + (0.310 − 0.538i)13-s + (−0.433 − 0.558i)14-s + (−0.125 + 0.216i)16-s + (0.257 + 0.445i)19-s + (0.452 − 0.783i)22-s + 0.259·23-s − 25-s + (−0.219 − 0.380i)26-s + (−0.495 + 0.0677i)28-s + (0.393 + 0.682i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.033803902\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.033803902\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + (-1.12 + 1.94i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 1.94i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + (-2.12 - 3.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.62 + 8.00i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.74 + 9.94i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.24 + 9.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.37 + 4.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.12 - 3.67i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 1.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.121 + 0.210i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.12 - 14.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.74 - 9.94i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.24 + 3.88i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.706084780921555509583976216765, −8.933195111248893071095787755303, −7.923409333253073023356702622372, −7.07050301650544773169553773833, −6.09638356597467066913215482113, −5.17306533412580048168666889105, −3.99226403441310973087061179695, −3.61401955531830710289242884803, −2.01722857757644649047580231820, −0.875607131160451955133827264559,
1.58069310904531437747395411289, 2.97887351528297223984012302770, 4.11800526393798678339728959162, 4.93179569527938255578046857822, 5.98922418081981259733355221532, 6.52406943404557237396144454896, 7.53624803431177605232277384990, 8.431750585775864112319648987018, 9.137850333454099666841649611625, 9.704038184873077430596540722838