| L(s) = 1 | + (2 − i)5-s + 4i·7-s + 4·11-s + 4i·13-s − 6i·17-s + 4·19-s − 4i·23-s + (3 − 4i)25-s + 4·29-s + (4 + 8i)35-s − 4i·37-s − 8·41-s + 12i·47-s − 9·49-s − 2i·53-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)5-s + 1.51i·7-s + 1.20·11-s + 1.10i·13-s − 1.45i·17-s + 0.917·19-s − 0.834i·23-s + (0.600 − 0.800i)25-s + 0.742·29-s + (0.676 + 1.35i)35-s − 0.657i·37-s − 1.24·41-s + 1.75i·47-s − 1.28·49-s − 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.491825161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491825161\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.829575609809099927549616554996, −8.490297534402433276537211892736, −7.06789268693692973831890982023, −6.53519549069998585644786240805, −5.71528983848918644149208669405, −5.07354328575928649516733747845, −4.26810741495646295736502979548, −2.90955782878987105188415109424, −2.19328313748489042836202646523, −1.15237084760825592137910308431,
0.972136604465642133399659551095, 1.78332920920835719852468907493, 3.32604611750637173917580441461, 3.69898209957509240611984397351, 4.85726124351368323471911354036, 5.75146765619206701106438276300, 6.53410898244224909583266021305, 7.07951219244992149513245311167, 7.899169213253494066929798070210, 8.703181825472352089130206028342
