| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 2·13-s + 14-s − 16-s − 3·17-s + 3·19-s − 23-s − 2·26-s − 28-s + 6·29-s + 2·31-s + 5·32-s − 3·34-s − 3·37-s + 3·38-s + 3·41-s − 12·43-s − 46-s + 47-s − 6·49-s + 2·52-s + 6·53-s − 3·56-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s + 0.688·19-s − 0.208·23-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.359·31-s + 0.883·32-s − 0.514·34-s − 0.493·37-s + 0.486·38-s + 0.468·41-s − 1.82·43-s − 0.147·46-s + 0.145·47-s − 6/7·49-s + 0.277·52-s + 0.824·53-s − 0.400·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
| 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
−15.43877652895571, −14.69652718428498, −14.58914882996825, −13.82556606888042, −13.46958475786243, −13.01393799190335, −12.27689634305106, −11.90973530650835, −11.45765954278035, −10.69750642556191, −9.983616473313691, −9.661782607452710, −8.892692638440453, −8.400509925580081, −7.939708550455215, −6.998948814275623, −6.620599798679025, −5.798242891583582, −5.224270546028373, −4.730462040032714, −4.220081198180473, −3.449279619142222, −2.827857202321235, −2.053074472557207, −1.002031089540292, 0,
1.002031089540292, 2.053074472557207, 2.827857202321235, 3.449279619142222, 4.220081198180473, 4.730462040032714, 5.224270546028373, 5.798242891583582, 6.620599798679025, 6.998948814275623, 7.939708550455215, 8.400509925580081, 8.892692638440453, 9.661782607452710, 9.983616473313691, 10.69750642556191, 11.45765954278035, 11.90973530650835, 12.27689634305106, 13.01393799190335, 13.46958475786243, 13.82556606888042, 14.58914882996825, 14.69652718428498, 15.43877652895571
