| L(s) = 1 | − 2·5-s + 7-s − 2·13-s + 17-s + 6·19-s − 8·23-s − 25-s + 4·29-s − 2·35-s + 10·37-s − 10·41-s − 8·43-s + 8·47-s + 49-s − 6·53-s + 4·59-s − 8·61-s + 4·65-s − 4·67-s + 8·71-s + 4·73-s − 2·85-s − 10·89-s − 2·91-s − 12·95-s − 18·101-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.554·13-s + 0.242·17-s + 1.37·19-s − 1.66·23-s − 1/5·25-s + 0.742·29-s − 0.338·35-s + 1.64·37-s − 1.56·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 1.02·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s + 0.468·73-s − 0.216·85-s − 1.05·89-s − 0.209·91-s − 1.23·95-s − 1.79·101-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−7.953214926954187995816044225871, −7.50771857506331378376495110708, −6.65298695003059198257226642180, −5.74848146635077068277698991513, −4.97278307510275160646233582453, −4.20602118699970139626930363628, −3.46704732461879922705217223274, −2.51906405567727999633690617948, −1.33244702604165751838533247806, 0,
1.33244702604165751838533247806, 2.51906405567727999633690617948, 3.46704732461879922705217223274, 4.20602118699970139626930363628, 4.97278307510275160646233582453, 5.74848146635077068277698991513, 6.65298695003059198257226642180, 7.50771857506331378376495110708, 7.953214926954187995816044225871
