| L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·11-s − 16-s + 7·17-s − 7·19-s − 3·22-s + 6·23-s − 5·25-s + 5·29-s + 5·32-s + 7·34-s − 8·37-s − 7·38-s + 2·43-s + 3·44-s + 6·46-s − 7·47-s − 5·50-s + 3·53-s + 5·58-s + 7·59-s + 7·61-s + 7·64-s + 3·67-s − 7·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.904·11-s − 1/4·16-s + 1.69·17-s − 1.60·19-s − 0.639·22-s + 1.25·23-s − 25-s + 0.928·29-s + 0.883·32-s + 1.20·34-s − 1.31·37-s − 1.13·38-s + 0.304·43-s + 0.452·44-s + 0.884·46-s − 1.02·47-s − 0.707·50-s + 0.412·53-s + 0.656·58-s + 0.911·59-s + 0.896·61-s + 7/8·64-s + 0.366·67-s − 0.848·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−14.41734481306169, −13.66997055060722, −13.40590805180104, −12.85354925118412, −12.36632532012687, −12.14590681500349, −11.39417046723268, −10.80652721162475, −10.26963285092950, −9.860049974812293, −9.320618793017554, −8.599910514831736, −8.223842323772323, −7.842311625196703, −6.905090032902731, −6.591647470778035, −5.739994205941476, −5.342649894806084, −5.030335940981522, −4.186522768883699, −3.824185503634710, −3.051175227436095, −2.663254277148751, −1.752467191391455, −0.8190270097980625, 0,
0.8190270097980625, 1.752467191391455, 2.663254277148751, 3.051175227436095, 3.824185503634710, 4.186522768883699, 5.030335940981522, 5.342649894806084, 5.739994205941476, 6.591647470778035, 6.905090032902731, 7.842311625196703, 8.223842323772323, 8.599910514831736, 9.320618793017554, 9.860049974812293, 10.26963285092950, 10.80652721162475, 11.39417046723268, 12.14590681500349, 12.36632532012687, 12.85354925118412, 13.40590805180104, 13.66997055060722, 14.41734481306169
