| L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s + 2·13-s − 17-s + 4·19-s − 2·21-s − 6·23-s − 5·25-s + 27-s − 5·29-s − 3·31-s + 33-s + 3·37-s + 2·39-s + 41-s + 7·43-s − 3·47-s − 3·49-s − 51-s − 10·53-s + 4·57-s − 61-s − 2·63-s − 2·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s + 0.192·27-s − 0.928·29-s − 0.538·31-s + 0.174·33-s + 0.493·37-s + 0.320·39-s + 0.156·41-s + 1.06·43-s − 0.437·47-s − 3/7·49-s − 0.140·51-s − 1.37·53-s + 0.529·57-s − 0.128·61-s − 0.251·63-s − 0.244·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.66980428807115231479937103006, −6.82859905209233024790970758309, −6.10012211567341971842158532907, −5.60099490110489015753775508014, −4.46819991018812124205077422334, −3.77053973020507729021356379144, −3.23751380039748455336285563168, −2.26197553158317614556998571042, −1.39459809147563263209556289118, 0,
1.39459809147563263209556289118, 2.26197553158317614556998571042, 3.23751380039748455336285563168, 3.77053973020507729021356379144, 4.46819991018812124205077422334, 5.60099490110489015753775508014, 6.10012211567341971842158532907, 6.82859905209233024790970758309, 7.66980428807115231479937103006
