| L(s) = 1 | − 3-s − 5·7-s + 9-s − 11-s + 2·13-s + 17-s − 19-s + 5·21-s + 4·23-s − 27-s + 9·29-s − 6·31-s + 33-s + 10·37-s − 2·39-s − 12·41-s + 5·43-s + 3·47-s + 18·49-s − 51-s + 12·53-s + 57-s − 4·59-s + 3·61-s − 5·63-s + 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.242·17-s − 0.229·19-s + 1.09·21-s + 0.834·23-s − 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.174·33-s + 1.64·37-s − 0.320·39-s − 1.87·41-s + 0.762·43-s + 0.437·47-s + 18/7·49-s − 0.140·51-s + 1.64·53-s + 0.132·57-s − 0.520·59-s + 0.384·61-s − 0.629·63-s + 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−12.78343667148374, −12.19265114201494, −11.92040409156769, −11.35861381265332, −10.68556048016341, −10.53217880531827, −9.946168743465377, −9.703332149022692, −9.014528249806666, −8.790147195585487, −8.165498148382002, −7.515435366646116, −6.968193936927838, −6.744628012832939, −6.216518941509644, −5.772088174521759, −5.428498429723309, −4.726449602932411, −4.128465477146632, −3.740654192594262, −2.964160191521997, −2.880672905438877, −2.067915045853990, −1.151005551898748, −0.6864695133126996, 0,
0.6864695133126996, 1.151005551898748, 2.067915045853990, 2.880672905438877, 2.964160191521997, 3.740654192594262, 4.128465477146632, 4.726449602932411, 5.428498429723309, 5.772088174521759, 6.216518941509644, 6.744628012832939, 6.968193936927838, 7.515435366646116, 8.165498148382002, 8.790147195585487, 9.014528249806666, 9.703332149022692, 9.946168743465377, 10.53217880531827, 10.68556048016341, 11.35861381265332, 11.92040409156769, 12.19265114201494, 12.78343667148374
