| L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 4·11-s + 13-s + 2·17-s − 4·19-s + 4·23-s − 25-s − 2·29-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 6·45-s + 8·47-s + 49-s − 2·53-s − 8·55-s + 4·59-s + 14·61-s − 3·63-s − 2·65-s − 12·67-s − 8·71-s + 14·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.07·55-s + 0.520·59-s + 1.79·61-s − 0.377·63-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.63·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.463225273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463225273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.709404849513276317953776452768, −8.101893458522369479762886726262, −7.32453385103454370756706987233, −6.49156772710404097717968435691, −5.75127750674865168649545369665, −4.79464304823446706064156184893, −3.92052118759426629531430503113, −3.30789383012068572342257842970, −2.07123031398452708976383958408, −0.74445041737067892319726100878,
0.74445041737067892319726100878, 2.07123031398452708976383958408, 3.30789383012068572342257842970, 3.92052118759426629531430503113, 4.79464304823446706064156184893, 5.75127750674865168649545369665, 6.49156772710404097717968435691, 7.32453385103454370756706987233, 8.101893458522369479762886726262, 8.709404849513276317953776452768
