| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s − 4·13-s + 16-s + 2·17-s + 2·19-s − 22-s + 2·23-s − 5·25-s + 4·26-s − 10·29-s + 8·31-s − 32-s − 2·34-s + 37-s − 2·38-s − 6·41-s + 6·43-s + 44-s − 2·46-s − 4·47-s + 5·50-s − 4·52-s + 6·53-s + 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.213·22-s + 0.417·23-s − 25-s + 0.784·26-s − 1.85·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.164·37-s − 0.324·38-s − 0.937·41-s + 0.914·43-s + 0.150·44-s − 0.294·46-s − 0.583·47-s + 0.707·50-s − 0.554·52-s + 0.824·53-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 358974 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 358974 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−12.58694084381056, −12.24388752245678, −11.76514258062037, −11.34569561941591, −11.06329562029388, −10.26096138687970, −9.958774941923068, −9.674137906824681, −9.151523584459012, −8.747735268553521, −8.087367225063197, −7.734502995367077, −7.307564165957297, −6.941758456751002, −6.289943025049257, −5.805358249363188, −5.377175106863631, −4.754614009368188, −4.290898403794794, −3.478394888159122, −3.243892876521462, −2.378932821971357, −2.066665228152993, −1.346011071512097, −0.6984598894429709, 0,
0.6984598894429709, 1.346011071512097, 2.066665228152993, 2.378932821971357, 3.243892876521462, 3.478394888159122, 4.290898403794794, 4.754614009368188, 5.377175106863631, 5.805358249363188, 6.289943025049257, 6.941758456751002, 7.307564165957297, 7.734502995367077, 8.087367225063197, 8.747735268553521, 9.151523584459012, 9.674137906824681, 9.958774941923068, 10.26096138687970, 11.06329562029388, 11.34569561941591, 11.76514258062037, 12.24388752245678, 12.58694084381056
