| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 11-s − 2·12-s + 2·13-s + 16-s − 6·17-s − 18-s + 4·19-s − 22-s + 2·23-s + 2·24-s − 2·26-s + 4·27-s − 10·29-s − 8·31-s − 32-s − 2·33-s + 6·34-s + 36-s − 8·37-s − 4·38-s − 4·39-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 + 0.301·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.213·22-s + 0.417·23-s + 0.408·24-s − 0.392·26-s + 0.769·27-s − 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + 1.02·34-s + 1/6·36-s − 1.31·37-s − 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−10.63386564786085326074357105776, −9.381879522590044261784002659703, −8.811471475246278252438242664394, −7.53533584933465565840485813225, −6.69634210067912510539169096291, −5.86401879992220243201868389926, −4.93146063967320632748167609932, −3.46929385258852245350501561594, −1.69722610211157075824716815435, 0,
1.69722610211157075824716815435, 3.46929385258852245350501561594, 4.93146063967320632748167609932, 5.86401879992220243201868389926, 6.69634210067912510539169096291, 7.53533584933465565840485813225, 8.811471475246278252438242664394, 9.381879522590044261784002659703, 10.63386564786085326074357105776
