| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 7-s + 8-s + 6·9-s − 11-s − 3·12-s − 14-s + 16-s − 5·17-s + 6·18-s − 7·19-s + 3·21-s − 22-s − 8·23-s − 3·24-s − 9·27-s − 28-s + 3·29-s − 5·31-s + 32-s + 3·33-s − 5·34-s + 6·36-s − 37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 0.301·11-s − 0.866·12-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 1.41·18-s − 1.60·19-s + 0.654·21-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 1.73·27-s − 0.188·28-s + 0.557·29-s − 0.898·31-s + 0.176·32-s + 0.522·33-s − 0.857·34-s + 36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 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
−10.63395836942048087882037196254, −9.949177863209794004022035401564, −8.500433692387952608359829976927, −7.13910609406921513970960332408, −6.38067572656577564407783222886, −5.80649463440223025002162889007, −4.73853812405040226503164718551, −3.98662044483560965500927790599, −2.08731553991579213664182261622, 0,
2.08731553991579213664182261622, 3.98662044483560965500927790599, 4.73853812405040226503164718551, 5.80649463440223025002162889007, 6.38067572656577564407783222886, 7.13910609406921513970960332408, 8.500433692387952608359829976927, 9.949177863209794004022035401564, 10.63395836942048087882037196254
