| L(s) = 1 | − i·2-s + 1.41i·3-s − 4-s + 1.41·6-s + i·8-s + 0.999·9-s − 2·11-s − 1.41i·12-s + 16-s − 1.41i·17-s − 0.999i·18-s − 7.07·19-s + 2i·22-s − 4i·23-s − 1.41·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.816i·3-s − 0.5·4-s + 0.577·6-s + 0.353i·8-s + 0.333·9-s − 0.603·11-s − 0.408i·12-s + 0.250·16-s − 0.342i·17-s − 0.235i·18-s − 1.62·19-s + 0.426i·22-s − 0.834i·23-s − 0.288·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4846293258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4846293258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 1.41iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89iT - 97T^{2} \) |
| 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
−9.021842192411181184677811653073, −7.973383052261569585452206659016, −7.20401527229691879488167052103, −6.10134708326366342986147386797, −5.22530009625841453639811876980, −4.37959716935488041952990505160, −3.88394352365241424384130695459, −2.75728794516669590052866321641, −1.83587082949540922336602752170, −0.15958919310640893485893370781,
1.39622223497822644934781913857, 2.42948268927100155239251657829, 3.77577075677783212847261880262, 4.58343787011781304545844076685, 5.61250838183266699608775584898, 6.25870595984772237712638429253, 7.06355564684671132370501797349, 7.60658621434077415217079308037, 8.302011868498730335271016441189, 9.020633332025144560865510053422
