L(s) = 1 | + (−2.70 + 1.56i)2-s + (−1.5 + 0.866i)3-s + (2.88 − 4.99i)4-s − 2.70·5-s + (2.70 − 4.68i)6-s + (−3.66 − 5.96i)7-s + 5.54i·8-s + (1.5 − 2.59i)9-s + (7.31 − 4.22i)10-s + (−5.80 + 3.35i)11-s + 9.99i·12-s + (12.6 + 3.01i)13-s + (19.2 + 10.4i)14-s + (4.05 − 2.34i)15-s + (2.88 + 4.99i)16-s + (−11.9 − 6.88i)17-s + ⋯ |
L(s) = 1 | + (−1.35 + 0.781i)2-s + (−0.5 + 0.288i)3-s + (0.721 − 1.24i)4-s − 0.540·5-s + (0.451 − 0.781i)6-s + (−0.522 − 0.852i)7-s + 0.692i·8-s + (0.166 − 0.288i)9-s + (0.731 − 0.422i)10-s + (−0.527 + 0.304i)11-s + 0.833i·12-s + (0.972 + 0.231i)13-s + (1.37 + 0.745i)14-s + (0.270 − 0.156i)15-s + (0.180 + 0.312i)16-s + (−0.701 − 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.238762 + 0.325064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.238762 + 0.325064i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (3.66 + 5.96i)T \) |
| 13 | \( 1 + (-12.6 - 3.01i)T \) |
good | 2 | \( 1 + (2.70 - 1.56i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + 2.70T + 25T^{2} \) |
| 11 | \( 1 + (5.80 - 3.35i)T + (60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (11.9 + 6.88i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.1 + 24.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-19.5 - 33.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-9.25 - 16.0i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 2.09T + 961T^{2} \) |
| 37 | \( 1 + (27.3 - 15.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-22.7 - 39.4i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (17.1 - 29.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 - 6.43T + 2.20e3T^{2} \) |
| 53 | \( 1 + 74.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-8.75 + 15.1i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-70.7 - 40.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-9.40 + 5.43i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-102. - 59.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 - 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 147.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (65.1 + 112. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (46.2 - 80.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.38441804000999470360685792975, −10.92436115083277766744069031565, −9.785550733360711877439165278875, −9.207836887788299066474252595739, −8.001622621328513541176551119491, −7.13701484244918961710015987512, −6.47524868122898325732794679241, −5.00538221149264731997309118282, −3.55194119913617959408567157471, −0.932651043722079011735844134127,
0.46631627231935247303772900059, 2.14442458637591431446094510124, 3.52477499707407095333375046427, 5.45568410604339088093338851932, 6.58577068826130741466754375913, 7.981783538696937120533120404865, 8.506811098919447831349561459986, 9.548206085897652165515903849022, 10.59655522892242261107228322438, 11.14491579951644206286152913795