| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 7-s + 8-s + 9-s + 10-s − 11-s + 2·12-s − 2·13-s + 14-s + 2·15-s + 16-s − 3·17-s + 18-s + 20-s + 2·21-s − 22-s + 2·24-s + 25-s − 2·26-s − 4·27-s + 28-s − 3·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.223·20-s + 0.436·21-s − 0.213·22-s + 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s + 0.188·28-s − 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.05676271454806, −12.57854959605685, −12.38815640405556, −11.54146018785318, −11.10645381187559, −10.90007240934838, −10.15349408568356, −9.703581304130695, −9.180205452049393, −8.953081311007244, −8.178096537888850, −7.943964844585924, −7.389554764297701, −6.921651197440620, −6.413869174666703, −5.681114962230454, −5.422737643916791, −4.787926992979624, −4.293384198274596, −3.698570524262465, −3.293860938959796, −2.563082582259070, −2.229910717371355, −1.878338478460743, −0.9988800890495596, 0,
0.9988800890495596, 1.878338478460743, 2.229910717371355, 2.563082582259070, 3.293860938959796, 3.698570524262465, 4.293384198274596, 4.787926992979624, 5.422737643916791, 5.681114962230454, 6.413869174666703, 6.921651197440620, 7.389554764297701, 7.943964844585924, 8.178096537888850, 8.953081311007244, 9.180205452049393, 9.703581304130695, 10.15349408568356, 10.90007240934838, 11.10645381187559, 11.54146018785318, 12.38815640405556, 12.57854959605685, 13.05676271454806
