| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s + 4·7-s − 8-s + 9-s + 10-s + 6·11-s − 2·12-s + 13-s − 4·14-s + 2·15-s + 16-s + 6·17-s − 18-s − 2·19-s − 20-s − 8·21-s − 6·22-s + 2·24-s + 25-s − 26-s + 4·27-s + 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.577·12-s + 0.277·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 1.74·21-s − 1.27·22-s + 0.408·24-s + 1/5·25-s − 0.196·26-s + 0.769·27-s + 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.53797733049997, −14.20087547812443, −13.38282134489259, −12.55449353664022, −12.13688049972619, −11.65418402290907, −11.39201646236745, −11.17961702729087, −10.37171546630979, −10.03870370450586, −9.262665676798819, −8.742695632255007, −8.261776772773876, −7.797988274805143, −7.184200784154998, −6.600837216930184, −6.153683097153443, −5.523691675323631, −4.987244585483402, −4.443291821602583, −3.709068093064503, −3.190735633601786, −1.940161210567461, −1.421863618139079, −0.9861501043336065, 0,
0.9861501043336065, 1.421863618139079, 1.940161210567461, 3.190735633601786, 3.709068093064503, 4.443291821602583, 4.987244585483402, 5.523691675323631, 6.153683097153443, 6.600837216930184, 7.184200784154998, 7.797988274805143, 8.261776772773876, 8.742695632255007, 9.262665676798819, 10.03870370450586, 10.37171546630979, 11.17961702729087, 11.39201646236745, 11.65418402290907, 12.13688049972619, 12.55449353664022, 13.38282134489259, 14.20087547812443, 14.53797733049997
