| L(s) = 1 | + 3-s − 4·7-s + 9-s − 4·11-s − 2·17-s − 19-s − 4·21-s + 2·23-s − 5·25-s + 27-s − 6·29-s − 6·31-s − 4·33-s − 8·37-s + 10·41-s + 12·43-s − 10·47-s + 9·49-s − 2·51-s + 2·53-s − 57-s − 4·59-s − 10·61-s − 4·63-s + 2·69-s + 16·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.485·17-s − 0.229·19-s − 0.872·21-s + 0.417·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.696·33-s − 1.31·37-s + 1.56·41-s + 1.82·43-s − 1.45·47-s + 9/7·49-s − 0.280·51-s + 0.274·53-s − 0.132·57-s − 0.520·59-s − 1.28·61-s − 0.503·63-s + 0.240·69-s + 1.89·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−9.482793580868359991017773314605, −9.097240421060539335944630760653, −7.898686491096330582694114446456, −7.23982863334654173329136409609, −6.24883441356449067760906148295, −5.36808784502647146902895503090, −4.02602094871991354322845522422, −3.15184971185467977468154125862, −2.18943331110765256094544725740, 0,
2.18943331110765256094544725740, 3.15184971185467977468154125862, 4.02602094871991354322845522422, 5.36808784502647146902895503090, 6.24883441356449067760906148295, 7.23982863334654173329136409609, 7.898686491096330582694114446456, 9.097240421060539335944630760653, 9.482793580868359991017773314605
