| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 5·7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 5·14-s − 15-s + 16-s − 4·17-s + 18-s + 5·19-s + 20-s + 5·21-s − 22-s + 6·23-s − 24-s + 25-s − 27-s − 5·28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 1.09·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.944·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.72222653583909, −13.78042683257993, −13.58994294952296, −13.11571750744722, −12.77930198411010, −12.07208188749433, −11.91546297257976, −10.88588160009421, −10.74601388262854, −10.09385223193053, −9.514674172568431, −9.178264090560683, −8.539350916180102, −7.513432757827069, −7.027545348349783, −6.670625372413728, −6.187704845029399, −5.524532763486534, −5.187249579869771, −4.469555926359485, −3.664048035964282, −3.191945441876394, −2.680623313526125, −1.847411561149116, −0.8732993190506238, 0,
0.8732993190506238, 1.847411561149116, 2.680623313526125, 3.191945441876394, 3.664048035964282, 4.469555926359485, 5.187249579869771, 5.524532763486534, 6.187704845029399, 6.670625372413728, 7.027545348349783, 7.513432757827069, 8.539350916180102, 9.178264090560683, 9.514674172568431, 10.09385223193053, 10.74601388262854, 10.88588160009421, 11.91546297257976, 12.07208188749433, 12.77930198411010, 13.11571750744722, 13.58994294952296, 13.78042683257993, 14.72222653583909
