L(s) = 1 | + (2.61 − 1.51i)2-s + (1.89 − 4.83i)3-s + (0.575 − 0.997i)4-s + (−18.8 − 10.8i)5-s + (−2.34 − 15.5i)6-s + (−0.934 + 0.539i)7-s + 20.7i·8-s + (−19.7 − 18.3i)9-s − 65.7·10-s + (25.6 − 14.8i)11-s + (−3.72 − 4.67i)12-s + (−35.7 − 30.2i)13-s + (−1.63 + 2.82i)14-s + (−88.2 + 70.3i)15-s + (35.9 + 62.2i)16-s + 85.5·17-s + ⋯ |
L(s) = 1 | + (0.926 − 0.534i)2-s + (0.365 − 0.930i)3-s + (0.0719 − 0.124i)4-s + (−1.68 − 0.971i)5-s + (−0.159 − 1.05i)6-s + (−0.0504 + 0.0291i)7-s + 0.915i·8-s + (−0.732 − 0.680i)9-s − 2.07·10-s + (0.704 − 0.406i)11-s + (−0.0897 − 0.112i)12-s + (−0.763 − 0.646i)13-s + (−0.0311 + 0.0539i)14-s + (−1.51 + 1.21i)15-s + (0.561 + 0.972i)16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.250443 - 1.68800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.250443 - 1.68800i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.89 + 4.83i)T \) |
| 13 | \( 1 + (35.7 + 30.2i)T \) |
good | 2 | \( 1 + (-2.61 + 1.51i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (18.8 + 10.8i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (0.934 - 0.539i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-25.6 + 14.8i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 - 85.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-79.6 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (35.2 + 61.0i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (181. + 105. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 112. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-73.5 - 42.4i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (36.5 + 63.2i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-194. + 112. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-148. - 85.9i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (22.9 + 39.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-778. - 449. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 569. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 148. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (434. + 752. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-440. + 254. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 448. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-245. + 142. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.57682169654398247954878561928, −11.93466866027230419721976954705, −11.22648150911310292694041001739, −8.968816439139063277080693169993, −8.163920012318790775804510818703, −7.25967220092020021755236651352, −5.32948361212517569163904678747, −4.06748905134871571497032726954, −2.95812989924302958477478681291, −0.67817881322242992704794447011,
3.45750226088733676996012938684, 3.98165622442987868330209668110, 5.28876817530858219069181804838, 6.95071592253874992852671709462, 7.75629327959000001856675255077, 9.375872349883638808809466034774, 10.42174352248988549822431637682, 11.59271278241640530239663678160, 12.41794307930961470978379088986, 14.19151244681069946098568858711