| L(s) = 1 | + 2·3-s + 7-s + 9-s − 3·11-s − 2·13-s + 17-s − 5·19-s + 2·21-s + 6·23-s − 4·27-s − 9·29-s + 5·31-s − 6·33-s − 2·37-s − 4·39-s + 3·41-s + 10·43-s + 49-s + 2·51-s + 12·53-s − 10·57-s − 9·59-s − 8·61-s + 63-s − 8·67-s + 12·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.242·17-s − 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.769·27-s − 1.67·29-s + 0.898·31-s − 1.04·33-s − 0.328·37-s − 0.640·39-s + 0.468·41-s + 1.52·43-s + 1/7·49-s + 0.280·51-s + 1.64·53-s − 1.32·57-s − 1.17·59-s − 1.02·61-s + 0.125·63-s − 0.977·67-s + 1.44·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.612352321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.612352321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.23296961485088, −12.71291184495727, −12.32702367846569, −11.73914770502828, −11.03598774236555, −10.79258735484543, −10.32446282276819, −9.658528257151821, −9.111474608581813, −8.989287949773452, −8.229312409034821, −7.982245053312798, −7.323860720828347, −7.228378655815290, −6.283269970911766, −5.784713119654126, −5.222324330483624, −4.702570530965590, −4.120209431767022, −3.586670128752834, −2.916726338386746, −2.476365246252096, −2.102759977792401, −1.319804094541945, −0.4089897767887607,
0.4089897767887607, 1.319804094541945, 2.102759977792401, 2.476365246252096, 2.916726338386746, 3.586670128752834, 4.120209431767022, 4.702570530965590, 5.222324330483624, 5.784713119654126, 6.283269970911766, 7.228378655815290, 7.323860720828347, 7.982245053312798, 8.229312409034821, 8.989287949773452, 9.111474608581813, 9.658528257151821, 10.32446282276819, 10.79258735484543, 11.03598774236555, 11.73914770502828, 12.32702367846569, 12.71291184495727, 13.23296961485088
