L(s) = 1 | + (1.22 + 1.22i)3-s + (−2.67 + 4.22i)5-s + (1.44 − 1.44i)7-s + 2.99i·9-s − 3.34·11-s + (10.4 + 10.4i)13-s + (−8.44 + 1.89i)15-s + (−2.65 + 2.65i)17-s + 20.6i·19-s + 3.55·21-s + (−16.4 − 16.4i)23-s + (−10.6 − 22.5i)25-s + (−3.67 + 3.67i)27-s + 0.853i·29-s + 18.6·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.534 + 0.844i)5-s + (0.207 − 0.207i)7-s + 0.333i·9-s − 0.304·11-s + (0.803 + 0.803i)13-s + (−0.563 + 0.126i)15-s + (−0.155 + 0.155i)17-s + 1.08i·19-s + 0.169·21-s + (−0.715 − 0.715i)23-s + (−0.427 − 0.903i)25-s + (−0.136 + 0.136i)27-s + 0.0294i·29-s + 0.603·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.198555922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.198555922\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (2.67 - 4.22i)T \) |
good | 7 | \( 1 + (-1.44 + 1.44i)T - 49iT^{2} \) |
| 11 | \( 1 + 3.34T + 121T^{2} \) |
| 13 | \( 1 + (-10.4 - 10.4i)T + 169iT^{2} \) |
| 17 | \( 1 + (2.65 - 2.65i)T - 289iT^{2} \) |
| 19 | \( 1 - 20.6iT - 361T^{2} \) |
| 23 | \( 1 + (16.4 + 16.4i)T + 529iT^{2} \) |
| 29 | \( 1 - 0.853iT - 841T^{2} \) |
| 31 | \( 1 - 18.6T + 961T^{2} \) |
| 37 | \( 1 + (38.0 - 38.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 28.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-22.4 - 22.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (19.7 - 19.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (28.6 + 28.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 111. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + (54.8 - 54.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 68T + 5.04e3T^{2} \) |
| 73 | \( 1 + (39.7 + 39.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 24.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (21.1 + 21.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 94.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-14.5 + 14.5i)T - 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32031914735698460223351662354, −9.450505636648255052100560412195, −8.301213904856245862185135056866, −7.952861719308570539244791794854, −6.77462273518579126287133823362, −6.09840022442191440083744706511, −4.68353950575694459936238701973, −3.88586543744074086443673984170, −3.03137006407431676539095151136, −1.73594999072043055050416648608,
0.35507209331814763957753984233, 1.62951601676445761546941401436, 2.98691107473498321156271461482, 4.01755931670868247766584076956, 5.08651322141869045918987793678, 5.91131289725027560395524690248, 7.14496429107860844329819725864, 7.87566347941421057022538654431, 8.622096016924822345693560062328, 9.144834689320957703977646558110