L(s) = 1 | + 1.41·2-s + 2.80i·3-s + 2.00·4-s + 2.23i·5-s + 3.96i·6-s + 2.82·8-s − 4.87·9-s + 3.16i·10-s + 5.61i·12-s − 6.27·15-s + 4.00·16-s − 6.89·18-s + 4.47i·20-s + 8.92·23-s + 7.93i·24-s − 5.00·25-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 1.61i·3-s + 1.00·4-s + 0.999i·5-s + 1.61i·6-s + 1.00·8-s − 1.62·9-s + 1.00i·10-s + 1.61i·12-s − 1.61·15-s + 1.00·16-s − 1.62·18-s + 1.00i·20-s + 1.86·23-s + 1.61i·24-s − 1.00·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23962 + 2.71342i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23962 + 2.71342i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.80iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 10.7T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 8.15iT - 41T^{2} \) |
| 43 | \( 1 - 3.62T + 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.219iT - 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 8.72iT - 83T^{2} \) |
| 89 | \( 1 - 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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
−10.53635153518643962175830187208, −9.726191508574479279156623191906, −8.878691709664543722622019653435, −7.53645337125957682401121316142, −6.76661251358561272762080543881, −5.63487639634591663130656292659, −5.06023315259356229678288856367, −3.90081929874144743661926817262, −3.43724645364158732843358271231, −2.36707736700809061146951364622,
1.04807716952604171679047883287, 1.99355681980930722526384021400, 3.18148592295115947273218545871, 4.53262469587491989664500153455, 5.47643410663086022802143310620, 6.18875015918778189987741489985, 7.16505494867290926015047692850, 7.68442203481587007100637814638, 8.601840731277728256309379369763, 9.578270930418631420326558228490