| L(s) = 1 | + 2·5-s − 7-s − 6·13-s + 17-s + 2·19-s − 25-s + 8·29-s − 2·35-s − 2·37-s − 2·41-s − 8·43-s + 8·47-s + 49-s − 2·53-s + 12·59-s + 4·61-s − 12·65-s − 12·67-s + 8·73-s − 8·79-s + 2·85-s − 10·89-s + 6·91-s + 4·95-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 1.66·13-s + 0.242·17-s + 0.458·19-s − 1/5·25-s + 1.48·29-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s + 0.512·61-s − 1.48·65-s − 1.46·67-s + 0.936·73-s − 0.900·79-s + 0.216·85-s − 1.05·89-s + 0.628·91-s + 0.410·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.33611694408721, −13.94983963682972, −13.43977332240790, −12.96264628665388, −12.38397067834188, −11.92160818381133, −11.62464667821861, −10.69137236763962, −10.15796232582074, −9.942312429373763, −9.490806012090595, −8.875222147570215, −8.309706617905440, −7.660265071325396, −7.077449218426652, −6.701087196873183, −6.019127484694627, −5.458850070847472, −5.004995344772440, −4.436932291301575, −3.628835965198323, −2.893417249311838, −2.447352438292791, −1.793746147514218, −0.9341015179824797, 0,
0.9341015179824797, 1.793746147514218, 2.447352438292791, 2.893417249311838, 3.628835965198323, 4.436932291301575, 5.004995344772440, 5.458850070847472, 6.019127484694627, 6.701087196873183, 7.077449218426652, 7.660265071325396, 8.309706617905440, 8.875222147570215, 9.490806012090595, 9.942312429373763, 10.15796232582074, 10.69137236763962, 11.62464667821861, 11.92160818381133, 12.38397067834188, 12.96264628665388, 13.43977332240790, 13.94983963682972, 14.33611694408721
