| L(s) = 1 | + 3-s + 2.37·5-s + 2.52i·7-s + 9-s + 2.52i·11-s − 1.58i·13-s + 2.37·15-s − 0.372·17-s + (4 + 1.73i)19-s + 2.52i·21-s + 1.87i·23-s + 0.627·25-s + 27-s + 3.16i·29-s − 2.74·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.06·5-s + 0.954i·7-s + 0.333·9-s + 0.761i·11-s − 0.439i·13-s + 0.612·15-s − 0.0902·17-s + (0.917 + 0.397i)19-s + 0.550i·21-s + 0.391i·23-s + 0.125·25-s + 0.192·27-s + 0.588i·29-s − 0.492·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19997 + 0.727289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19997 + 0.727289i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
| good | 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 23 | \( 1 - 1.87iT - 23T^{2} \) |
| 29 | \( 1 - 3.16iT - 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 1.58iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 0.644iT - 43T^{2} \) |
| 47 | \( 1 - 0.939iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.372T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
| 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
−9.864491064924430720535305363000, −9.453962770426214801266573518207, −8.659613163180345964400072195597, −7.70303047445539108235159024637, −6.78216399252225813885329820980, −5.67341884168134899915176535878, −5.15326131536363447616618362452, −3.67597209665822873604286440429, −2.52634863201967623968541861589, −1.70162371498176958513907432921,
1.15485181654875614049428222535, 2.45199463372043304074658687339, 3.55220509058854867646776787379, 4.61260135922889494099416621981, 5.73396342925952404787992959077, 6.60983858287469009027381671001, 7.47816862044829957790709997018, 8.379571908016071201627880174985, 9.345985989364276736158634488970, 9.830931970307129000636337999682
