| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s + 11-s − 2·13-s − 3·14-s + 16-s + 5·17-s + 3·19-s + 22-s − 5·23-s − 2·26-s − 3·28-s + 29-s + 32-s + 5·34-s − 3·37-s + 3·38-s + 3·41-s + 4·43-s + 44-s − 5·46-s + 13·47-s + 2·49-s − 2·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s − 0.801·14-s + 1/4·16-s + 1.21·17-s + 0.688·19-s + 0.213·22-s − 1.04·23-s − 0.392·26-s − 0.566·28-s + 0.185·29-s + 0.176·32-s + 0.857·34-s − 0.493·37-s + 0.486·38-s + 0.468·41-s + 0.609·43-s + 0.150·44-s − 0.737·46-s + 1.89·47-s + 2/7·49-s − 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.78206755935411, −13.03884909926130, −12.63568658287643, −12.27398485686292, −11.89938747924929, −11.43178122619724, −10.70387461505623, −10.20210650590751, −9.879166474330746, −9.400251476123202, −8.879218670804949, −8.156815897074600, −7.590519764770322, −7.215525301759248, −6.694706651936748, −6.108529238669496, −5.547468312717486, −5.426165821993158, −4.356989814593530, −4.110515976990820, −3.410708794428661, −2.941222378189237, −2.475001032736807, −1.604279648615239, −0.9031769524699815, 0,
0.9031769524699815, 1.604279648615239, 2.475001032736807, 2.941222378189237, 3.410708794428661, 4.110515976990820, 4.356989814593530, 5.426165821993158, 5.547468312717486, 6.108529238669496, 6.694706651936748, 7.215525301759248, 7.590519764770322, 8.156815897074600, 8.879218670804949, 9.400251476123202, 9.879166474330746, 10.20210650590751, 10.70387461505623, 11.43178122619724, 11.89938747924929, 12.27398485686292, 12.63568658287643, 13.03884909926130, 13.78206755935411
