| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s + 4·11-s + 2·13-s + 15-s + 6·17-s − 8·19-s + 4·21-s + 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 4·35-s + 2·37-s + 2·39-s + 2·41-s − 43-s + 45-s − 8·47-s + 9·49-s + 6·51-s − 6·53-s + 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.518705958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.518705958\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−14.64182346648465, −14.29207651941656, −13.96110011288604, −13.22370398522865, −12.76571551701934, −12.05894238522926, −11.67726218123529, −11.04500422472872, −10.62889608371552, −9.888387924799280, −9.542910265266206, −8.592205112401612, −8.541333276054192, −7.944861430008348, −7.354012824925349, −6.520783798031062, −6.178539767055708, −5.432655208868023, −4.688870808034163, −4.264861140490469, −3.627802032258308, −2.848693651165031, −1.972773852349072, −1.544940106253870, −0.8899050530534210,
0.8899050530534210, 1.544940106253870, 1.972773852349072, 2.848693651165031, 3.627802032258308, 4.264861140490469, 4.688870808034163, 5.432655208868023, 6.178539767055708, 6.520783798031062, 7.354012824925349, 7.944861430008348, 8.541333276054192, 8.592205112401612, 9.542910265266206, 9.888387924799280, 10.62889608371552, 11.04500422472872, 11.67726218123529, 12.05894238522926, 12.76571551701934, 13.22370398522865, 13.96110011288604, 14.29207651941656, 14.64182346648465
