| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s + 4·13-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·24-s − 5·25-s − 4·26-s − 4·27-s − 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s − 2·37-s − 2·38-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.71688946650247, −14.20320025609894, −13.83501630963758, −13.31033589441407, −12.81926668702262, −12.01878168502246, −11.62333605457552, −11.17705205494114, −10.32275486653526, −10.03890120975865, −9.436254157215824, −9.022458906380164, −8.354753202648190, −8.110149332059267, −7.597697606542388, −7.010624023945231, −6.333816121314228, −5.571323083846240, −5.326160933068359, −3.974482093861297, −3.739191873824977, −3.068494989002011, −2.495778332910390, −1.637430993271355, −1.181738165349491, 0,
1.181738165349491, 1.637430993271355, 2.495778332910390, 3.068494989002011, 3.739191873824977, 3.974482093861297, 5.326160933068359, 5.571323083846240, 6.333816121314228, 7.010624023945231, 7.597697606542388, 8.110149332059267, 8.354753202648190, 9.022458906380164, 9.436254157215824, 10.03890120975865, 10.32275486653526, 11.17705205494114, 11.62333605457552, 12.01878168502246, 12.81926668702262, 13.31033589441407, 13.83501630963758, 14.20320025609894, 14.71688946650247
