| L(s) = 1 | + 3-s − 3.46i·7-s − 2·9-s − 3i·11-s + 3.46·13-s + 3i·17-s + i·19-s − 3.46i·21-s − 5·27-s − 10.3i·29-s + 6.92·31-s − 3i·33-s − 10.3·37-s + 3.46·39-s − 9·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.30i·7-s − 0.666·9-s − 0.904i·11-s + 0.960·13-s + 0.727i·17-s + 0.229i·19-s − 0.755i·21-s − 0.962·27-s − 1.92i·29-s + 1.24·31-s − 0.522i·33-s − 1.70·37-s + 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640207799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640207799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
| 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
−9.016639175720539602942443957766, −8.222144354163242430940097231865, −7.958005099769256605870595972922, −6.66808310543572713742591452838, −6.11317597043784557410466137198, −4.99176828358205324890775730627, −3.67330014444403732992126931491, −3.49680976537946342037117879445, −1.97888845704182359167537715346, −0.57711382498361898731068023806,
1.66144543216919699029578046431, 2.72011967522874503846856554811, 3.39795133486916988462435407747, 4.79148554203322882464100533833, 5.47527955356213289076509702999, 6.41351882844141578071343444662, 7.25897991099273156585395210824, 8.360549068463395476762320651591, 8.779687161606344732401016393216, 9.356009648349112538084930154133
