| L(s) = 1 | − 3-s − 7-s + 9-s + 2·13-s − 4·17-s − 19-s + 21-s + 4·23-s − 27-s + 4·29-s + 10·37-s − 2·39-s + 10·41-s + 4·43-s − 6·47-s + 49-s + 4·51-s + 57-s − 4·59-s − 10·61-s − 63-s − 4·67-s − 4·69-s + 2·71-s + 2·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.970·17-s − 0.229·19-s + 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.742·29-s + 1.64·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.560·51-s + 0.132·57-s − 0.520·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.481·69-s + 0.237·71-s + 0.234·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 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.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−13.27358742164380, −13.06450622227206, −12.65015251812614, −12.12320813482089, −11.44875428236766, −11.21179745254521, −10.71725945220185, −10.33670425040335, −9.573777829057358, −9.335617572033790, −8.719933961514985, −8.291331660937710, −7.522372454409460, −7.257846569680237, −6.458316112083410, −6.215662350889501, −5.812834639735684, −5.029981251449221, −4.467730618319607, −4.216673863196973, −3.382843617020688, −2.794205183461132, −2.262046519866433, −1.364353311428020, −0.8198334711691560, 0,
0.8198334711691560, 1.364353311428020, 2.262046519866433, 2.794205183461132, 3.382843617020688, 4.216673863196973, 4.467730618319607, 5.029981251449221, 5.812834639735684, 6.215662350889501, 6.458316112083410, 7.257846569680237, 7.522372454409460, 8.291331660937710, 8.719933961514985, 9.335617572033790, 9.573777829057358, 10.33670425040335, 10.71725945220185, 11.21179745254521, 11.44875428236766, 12.12320813482089, 12.65015251812614, 13.06450622227206, 13.27358742164380
