L(s) = 1 | + (−1.44 + 2.43i)2-s + (−3.82 − 7.02i)4-s + (−14.6 + 8.45i)5-s + (3.08 + 1.78i)7-s + (22.6 + 0.866i)8-s + (0.610 − 47.8i)10-s + (25.0 − 43.4i)11-s + (−18.9 − 32.9i)13-s + (−8.80 + 4.93i)14-s + (−34.7 + 53.7i)16-s − 84.3i·17-s − 62.9i·19-s + (115. + 70.6i)20-s + (69.3 + 123. i)22-s + (37.6 + 65.1i)23-s + ⋯ |
L(s) = 1 | + (−0.511 + 0.859i)2-s + (−0.477 − 0.878i)4-s + (−1.31 + 0.756i)5-s + (0.166 + 0.0963i)7-s + (0.999 + 0.0382i)8-s + (0.0193 − 1.51i)10-s + (0.687 − 1.19i)11-s + (−0.405 − 0.701i)13-s + (−0.168 + 0.0941i)14-s + (−0.543 + 0.839i)16-s − 1.20i·17-s − 0.759i·19-s + (1.29 + 0.789i)20-s + (0.671 + 1.19i)22-s + (0.340 + 0.590i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.474658 - 0.249675i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.474658 - 0.249675i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.44 - 2.43i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (14.6 - 8.45i)T + (62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-3.08 - 1.78i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-25.0 + 43.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (18.9 + 32.9i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 84.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 62.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-37.6 - 65.1i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (105. + 60.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-17.2 + 9.97i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 17.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + (299. - 172. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (113. + 65.3i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (153. - 265. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 479. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (245. + 425. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (49.9 - 86.4i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-536. + 309. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 100.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (856. + 494. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-251. + 436. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.01e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (503. - 872. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.37329512225632316127872196742, −11.59707907716973251122907305056, −11.08560540858152431068881843666, −9.648289011998771317705576281379, −8.427264016076640333973613524643, −7.53336737365387961450004924765, −6.57945742375748918096743242486, −5.04277420588708608018828167939, −3.38166810561192475358255757468, −0.37068278149258161712504420677,
1.59408566353163872504724116725, 3.83159952568253013993951015699, 4.61821992910703449994642824985, 7.10700104897495741547386225443, 8.148015512766316973238074476216, 9.052475332948067608269922211129, 10.23532243646694516509151609946, 11.46911022253242174786521600032, 12.24065967701762377968642398442, 12.78625039718952176915282430047