| L(s) = 1 | + 2·5-s + 2·7-s + 11-s + 17-s − 2·19-s − 6·23-s − 25-s + 2·29-s + 4·35-s − 2·37-s − 2·41-s − 10·43-s + 4·47-s − 3·49-s + 4·53-s + 2·55-s − 6·59-s + 10·61-s − 12·67-s + 6·71-s − 4·73-s + 2·77-s + 2·79-s + 12·83-s + 2·85-s + 2·89-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.301·11-s + 0.242·17-s − 0.458·19-s − 1.25·23-s − 1/5·25-s + 0.371·29-s + 0.676·35-s − 0.328·37-s − 0.312·41-s − 1.52·43-s + 0.583·47-s − 3/7·49-s + 0.549·53-s + 0.269·55-s − 0.781·59-s + 1.28·61-s − 1.46·67-s + 0.712·71-s − 0.468·73-s + 0.227·77-s + 0.225·79-s + 1.31·83-s + 0.216·85-s + 0.211·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−13.90419758351888, −13.47169693550280, −13.15145022543494, −12.26283591069066, −12.07700837771028, −11.54683435580008, −10.92594062677598, −10.48401262462609, −9.884126313407208, −9.708933507050988, −8.932233386323345, −8.456212315252080, −8.072802179132519, −7.445219066039910, −6.863110351884551, −6.207098527169001, −5.946000684614341, −5.225340291941425, −4.822652173257693, −4.141777827409410, −3.595071094469396, −2.852806161380242, −1.969360738997966, −1.863113280394123, −1.012957213913877, 0,
1.012957213913877, 1.863113280394123, 1.969360738997966, 2.852806161380242, 3.595071094469396, 4.141777827409410, 4.822652173257693, 5.225340291941425, 5.946000684614341, 6.207098527169001, 6.863110351884551, 7.445219066039910, 8.072802179132519, 8.456212315252080, 8.932233386323345, 9.708933507050988, 9.884126313407208, 10.48401262462609, 10.92594062677598, 11.54683435580008, 12.07700837771028, 12.26283591069066, 13.15145022543494, 13.47169693550280, 13.90419758351888
