L(s) = 1 | + 0.414i·2-s + 1.82·4-s − 2.41i·7-s + 1.58i·8-s + 11-s − 2.82i·13-s + 0.999·14-s + 3·16-s + 0.414i·17-s − 3.58·19-s + 0.414i·22-s + i·23-s + 1.17·26-s − 4.41i·28-s + 6.82·29-s + ⋯ |
L(s) = 1 | + 0.292i·2-s + 0.914·4-s − 0.912i·7-s + 0.560i·8-s + 0.301·11-s − 0.784i·13-s + 0.267·14-s + 0.750·16-s + 0.100i·17-s − 0.822·19-s + 0.0883i·22-s + 0.208i·23-s + 0.229·26-s − 0.834i·28-s + 1.26·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.336138276\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.336138276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 0.414iT - 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 5.82iT - 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 0.343iT - 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 + 3.65iT - 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.6iT - 67T^{2} \) |
| 71 | \( 1 + 2.17T + 71T^{2} \) |
| 73 | \( 1 - 3.17iT - 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.171iT - 97T^{2} \) |
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
−8.475265550529535371851710393437, −8.187151010416979929019807258735, −7.12735710776379555242053239453, −6.74469854820058251521346718012, −5.91941450071104346119706323586, −5.01424335622174041718690737792, −4.00282648596314592558570647686, −3.09824448593868953450048168934, −2.07220445527602218971514067218, −0.825837566133919069551858940146,
1.26639940084778147362999521422, 2.32890239594830230500865583367, 2.95275360860394555886894667332, 4.13315223288798006548651766528, 5.01311473730984656468624379926, 6.23191696830421030612315867783, 6.44295544707527440008838545362, 7.36427216382886390957057215070, 8.420146607192707520520418263320, 8.827696328395097008622043368916