L(s) = 1 | − i·5-s − 1.63i·7-s + 3.95·11-s − 0.585·13-s + 4i·17-s + 7.91i·19-s − 5.59·23-s − 25-s + 7.65i·29-s + 5.59i·31-s − 1.63·35-s + 9.07·37-s + 1.41i·41-s − 7.91i·43-s + 3.27·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 0.619i·7-s + 1.19·11-s − 0.162·13-s + 0.970i·17-s + 1.81i·19-s − 1.16·23-s − 0.200·25-s + 1.42i·29-s + 1.00i·31-s − 0.277·35-s + 1.49·37-s + 0.220i·41-s − 1.20i·43-s + 0.478·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.760332107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760332107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 1.63iT - 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 7.91iT - 19T^{2} \) |
| 23 | \( 1 + 5.59T + 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 - 5.59iT - 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 7.91iT - 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 - 5.17iT - 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 + 10.2iT - 79T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 - 5.41iT - 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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
−8.869202605903624043856446708479, −8.055180136658735949279571223515, −7.47275669297353105654942339396, −6.41622840118481888233655382187, −5.96144565435835448009776597054, −4.88487661890120544920768645039, −3.93910255020595304086284638360, −3.56390601710761963128041247114, −1.91561841288903355510489933925, −1.13849376961797709324512791744,
0.63575014393193569196652788980, 2.20848205121969443505050239236, 2.82093596657131920067120015442, 4.03894203887196505012707908061, 4.67699444635761927528198211698, 5.82387824911023386308755757974, 6.36414841646036761581620134905, 7.16773547473073688790229512129, 7.892345135368314978513005755132, 8.808015352539101119239057379826