| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 3·11-s + 2·12-s − 13-s + 16-s + 7·17-s − 18-s − 3·22-s − 4·23-s − 2·24-s − 5·25-s + 26-s − 4·27-s − 3·29-s − 3·31-s − 32-s + 6·33-s − 7·34-s + 36-s − 2·39-s − 10·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 0.277·13-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.639·22-s − 0.834·23-s − 0.408·24-s − 25-s + 0.196·26-s − 0.769·27-s − 0.557·29-s − 0.538·31-s − 0.176·32-s + 1.04·33-s − 1.20·34-s + 1/6·36-s − 0.320·39-s − 1.56·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.60813900470629436081590957221, −6.96037695904906590046849789480, −6.07134327053067925547020726212, −5.48666233684396559822943441992, −4.36651204663874425703938926936, −3.39725444899692812604505244126, −3.21339795383222764825468351311, −1.94667677366717441752939715034, −1.51679629617059751376945059021, 0,
1.51679629617059751376945059021, 1.94667677366717441752939715034, 3.21339795383222764825468351311, 3.39725444899692812604505244126, 4.36651204663874425703938926936, 5.48666233684396559822943441992, 6.07134327053067925547020726212, 6.96037695904906590046849789480, 7.60813900470629436081590957221
