| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s − 2·9-s + 2·10-s + 11-s + 2·12-s − 4·13-s − 15-s − 4·16-s + 2·17-s + 4·18-s − 2·20-s − 2·22-s − 23-s − 4·25-s + 8·26-s − 5·27-s + 2·30-s − 7·31-s + 8·32-s + 33-s − 4·34-s − 4·36-s + 3·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 2/3·9-s + 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s − 0.447·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s + 0.365·30-s − 1.25·31-s + 1.41·32-s + 0.174·33-s − 0.685·34-s − 2/3·36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−10.03359978338253814456351958606, −9.410787129321073286385580378997, −8.678914216536525529374637909269, −7.77729309499855305690824525396, −7.38142851192821190173908719125, −5.98958560322778025003846316474, −4.56768346065918811492983607197, −3.18179592020545258264561636452, −1.88740391319498359233714309471, 0,
1.88740391319498359233714309471, 3.18179592020545258264561636452, 4.56768346065918811492983607197, 5.98958560322778025003846316474, 7.38142851192821190173908719125, 7.77729309499855305690824525396, 8.678914216536525529374637909269, 9.410787129321073286385580378997, 10.03359978338253814456351958606
