| L(s) = 1 | + (−0.338 − 1.37i)2-s + (−2.58 − 1.87i)3-s + (−1.77 + 0.929i)4-s + (−0.602 − 2.15i)5-s + (−1.70 + 4.17i)6-s − 2.97i·7-s + (1.87 + 2.11i)8-s + (2.21 + 6.82i)9-s + (−2.75 + 1.55i)10-s + (−0.263 − 0.0855i)11-s + (6.31 + 0.922i)12-s + (1.36 + 4.19i)13-s + (−4.08 + 1.00i)14-s + (−2.48 + 6.68i)15-s + (2.27 − 3.29i)16-s + (−3.01 − 4.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.239 − 0.970i)2-s + (−1.49 − 1.08i)3-s + (−0.885 + 0.464i)4-s + (−0.269 − 0.963i)5-s + (−0.694 + 1.70i)6-s − 1.12i·7-s + (0.663 + 0.748i)8-s + (0.739 + 2.27i)9-s + (−0.870 + 0.491i)10-s + (−0.0794 − 0.0258i)11-s + (1.82 + 0.266i)12-s + (0.378 + 1.16i)13-s + (−1.09 + 0.269i)14-s + (−0.641 + 1.72i)15-s + (0.568 − 0.822i)16-s + (−0.730 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.203749 + 0.266429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203749 + 0.266429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.338 + 1.37i)T \) |
| 5 | \( 1 + (0.602 + 2.15i)T \) |
| good | 3 | \( 1 + (2.58 + 1.87i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 2.97iT - 7T^{2} \) |
| 11 | \( 1 + (0.263 + 0.0855i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 4.19i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.01 + 4.14i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.824 + 1.13i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.307 + 0.0998i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 3.56i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.04 - 2.21i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.24 - 3.84i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.97 + 6.09i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + (-1.67 + 2.30i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.378 - 0.274i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (5.09 - 1.65i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.96 + 0.638i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.25 + 0.908i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.81 - 5.67i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.54 + 2.77i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.49 - 3.26i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 7.80i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.992 + 3.05i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-9.69 + 13.3i)T + (-29.9 - 92.2i)T^{2} \) |
| 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
−11.68670831641202456971553123020, −11.19851070849075037439868072803, −10.15493562260191061438806554487, −8.828223924715491688626357734137, −7.61045285188056329727603617569, −6.68641288696171765626053564807, −5.08449720657082378939364412477, −4.27080607348761645707403057281, −1.66805980467646199897764483500, −0.39363889497392438896570177008,
3.69591254203828779449422480795, 5.04889049538137091786837689451, 5.95746085065396692742899893083, 6.54516226692436522837154425034, 8.114534900358007246602869784790, 9.306174874848362267356709832043, 10.39269058726818731879149190826, 10.85143776247858404019895817300, 12.00559981258494555581098240917, 13.02706920899985841641786904365
