| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 2·11-s − 13-s − 15-s − 2·17-s + 2·19-s − 2·21-s − 8·23-s + 25-s − 27-s − 6·29-s − 2·31-s − 2·33-s + 2·35-s + 2·37-s + 39-s − 2·41-s + 45-s + 6·47-s − 3·49-s + 2·51-s − 10·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.258·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.348·33-s + 0.338·35-s + 0.328·37-s + 0.160·39-s − 0.312·41-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−7.63812592346065427631134645534, −6.99936705819988579507261660136, −6.05502514516369207202824543114, −5.74918695424804295084918684776, −4.74831701842715043890210360189, −4.26824676151392340843652258466, −3.25066678092891094215388240738, −2.02676226080377374314207754321, −1.45640443763892834519468612302, 0,
1.45640443763892834519468612302, 2.02676226080377374314207754321, 3.25066678092891094215388240738, 4.26824676151392340843652258466, 4.74831701842715043890210360189, 5.74918695424804295084918684776, 6.05502514516369207202824543114, 6.99936705819988579507261660136, 7.63812592346065427631134645534
