| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s + 3·14-s − 15-s + 16-s − 18-s − 7·19-s − 20-s − 3·21-s + 2·22-s − 9·23-s − 24-s − 4·25-s + 27-s − 3·28-s − 4·29-s + 30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.654·21-s + 0.426·22-s − 1.87·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.566·28-s − 0.742·29-s + 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−12.82416561606666, −12.64094202696371, −11.99722678184430, −11.55571273052985, −10.96078646982402, −10.55015526206224, −10.06316807100026, −9.693448550957880, −9.369321373306639, −8.634631022817623, −8.386111197563839, −7.929545992421500, −7.423223070364608, −7.017878544862528, −6.312945395139479, −6.164284046584284, −5.439359427117937, −4.835800359053337, −3.918178235289879, −3.759078960120302, −3.339482889686285, −2.358622228080121, −2.196812900548934, −1.602469653534827, −0.4143138331321834, 0,
0.4143138331321834, 1.602469653534827, 2.196812900548934, 2.358622228080121, 3.339482889686285, 3.759078960120302, 3.918178235289879, 4.835800359053337, 5.439359427117937, 6.164284046584284, 6.312945395139479, 7.017878544862528, 7.423223070364608, 7.929545992421500, 8.386111197563839, 8.634631022817623, 9.369321373306639, 9.693448550957880, 10.06316807100026, 10.55015526206224, 10.96078646982402, 11.55571273052985, 11.99722678184430, 12.64094202696371, 12.82416561606666
