| L(s) = 1 | + (−1.36 + 0.366i)2-s + (−4.08 + 1.09i)3-s + (1.73 − i)4-s + (4.55 + 2.05i)5-s + (5.17 − 2.99i)6-s + (−1.49 + 5.58i)7-s + (−1.99 + 2i)8-s + (7.69 − 4.44i)9-s + (−6.97 − 1.13i)10-s + (−11.2 − 6.46i)11-s + (−5.98 + 5.98i)12-s + (−11.6 − 5.82i)13-s − 8.17i·14-s + (−20.8 − 3.40i)15-s + (1.99 − 3.46i)16-s + (−4.19 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 + 0.183i)2-s + (−1.36 + 0.364i)3-s + (0.433 − 0.250i)4-s + (0.911 + 0.410i)5-s + (0.863 − 0.498i)6-s + (−0.213 + 0.797i)7-s + (−0.249 + 0.250i)8-s + (0.854 − 0.493i)9-s + (−0.697 − 0.113i)10-s + (−1.01 − 0.587i)11-s + (−0.498 + 0.498i)12-s + (−0.894 − 0.448i)13-s − 0.583i·14-s + (−1.39 − 0.226i)15-s + (0.124 − 0.216i)16-s + (−0.246 − 0.0661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0233432 - 0.114030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0233432 - 0.114030i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 - 0.366i)T \) |
| 5 | \( 1 + (-4.55 - 2.05i)T \) |
| 13 | \( 1 + (11.6 + 5.82i)T \) |
| good | 3 | \( 1 + (4.08 - 1.09i)T + (7.79 - 4.5i)T^{2} \) |
| 7 | \( 1 + (1.49 - 5.58i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (11.2 + 6.46i)T + (60.5 + 104. i)T^{2} \) |
| 17 | \( 1 + (4.19 + 1.12i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (8.59 + 14.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (36.8 - 9.86i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (1.00 + 0.583i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 27.0iT - 961T^{2} \) |
| 37 | \( 1 + (-17.0 + 4.57i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (52.8 + 30.4i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (4.18 - 15.6i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (44.8 + 44.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (14.5 + 14.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-56.3 - 97.5i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21.3 - 37.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-89.2 + 23.9i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (60.1 - 34.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (26.2 - 26.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 77.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (75.7 - 75.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (87.0 - 150. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (9.33 - 34.8i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.54415705510138120375106627700, −12.35110201654912330208154314085, −11.36071267603211607565460499221, −10.40718470955892802304068057912, −9.845050095022588253455119745337, −8.485730488062836870996398662800, −6.90380692039967907399260310769, −5.81852261588226574223143036836, −5.22058847304892628266774231307, −2.50838681347945712217199178303,
0.10677180101305673689870727127, 1.95910884997840067744062392234, 4.65880354833516278135163932513, 5.94334943458918547604646375464, 6.89514034726231314547569149789, 8.106627213066661101856673084388, 9.902849447525468482113603327836, 10.17985372526255768696074320882, 11.39852034270388760338191254668, 12.44887172238293491144437767124
