| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 3·11-s − 6·13-s − 15-s + 4·17-s − 2·19-s − 2·21-s + 23-s + 25-s + 27-s + 29-s + 3·33-s + 2·35-s − 3·37-s − 6·39-s − 5·41-s − 43-s − 45-s + 2·47-s − 3·49-s + 4·51-s + 53-s − 3·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s − 0.458·19-s − 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s + 0.522·33-s + 0.338·35-s − 0.493·37-s − 0.960·39-s − 0.780·41-s − 0.152·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.560·51-s + 0.137·53-s − 0.404·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| 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
−8.208437996531495020373479818233, −7.41430900414930497647301928165, −6.92249087490641000206533907977, −6.08678678212460068677357007738, −5.02675017190925940882265645467, −4.26099377949053555684999232267, −3.37028702078793469876326589654, −2.73652512678418972473043268488, −1.52113777313121416861734275364, 0,
1.52113777313121416861734275364, 2.73652512678418972473043268488, 3.37028702078793469876326589654, 4.26099377949053555684999232267, 5.02675017190925940882265645467, 6.08678678212460068677357007738, 6.92249087490641000206533907977, 7.41430900414930497647301928165, 8.208437996531495020373479818233
