| L(s) = 1 | + 2-s + 4-s + 8-s + 3·11-s − 13-s + 16-s − 3·17-s + 5·19-s + 3·22-s − 6·23-s − 26-s + 6·29-s + 5·31-s + 32-s − 3·34-s − 8·37-s + 5·38-s − 6·41-s + 10·43-s + 3·44-s − 6·46-s − 52-s − 6·53-s + 6·58-s + 12·59-s + 11·61-s + 5·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.904·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.639·22-s − 1.25·23-s − 0.196·26-s + 1.11·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s − 1.31·37-s + 0.811·38-s − 0.937·41-s + 1.52·43-s + 0.452·44-s − 0.884·46-s − 0.138·52-s − 0.824·53-s + 0.787·58-s + 1.56·59-s + 1.40·61-s + 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−12.89797049305696, −12.47879399464193, −11.98971451885622, −11.67210410531485, −11.41026016724782, −10.67039040079851, −10.17510584001730, −9.898397651880735, −9.275700919702732, −8.799523318220502, −8.239315879388962, −7.870588993893455, −7.123419789660056, −6.808317926591456, −6.420012692780021, −5.800111658408310, −5.365007987088000, −4.774369136864801, −4.354789081200615, −3.745690725296660, −3.436267272192481, −2.582753455941114, −2.292526379690363, −1.451073475005336, −0.9669488011081494, 0,
0.9669488011081494, 1.451073475005336, 2.292526379690363, 2.582753455941114, 3.436267272192481, 3.745690725296660, 4.354789081200615, 4.774369136864801, 5.365007987088000, 5.800111658408310, 6.420012692780021, 6.808317926591456, 7.123419789660056, 7.870588993893455, 8.239315879388962, 8.799523318220502, 9.275700919702732, 9.898397651880735, 10.17510584001730, 10.67039040079851, 11.41026016724782, 11.67210410531485, 11.98971451885622, 12.47879399464193, 12.89797049305696
