| L(s) = 1 | − 2·3-s + 9-s + 2·11-s + 13-s − 2·17-s + 2·19-s + 2·23-s + 4·27-s + 6·29-s − 2·31-s − 4·33-s − 6·37-s − 2·39-s + 2·41-s − 6·43-s − 8·47-s − 7·49-s + 4·51-s − 2·53-s − 4·57-s + 6·59-s + 14·61-s − 4·69-s − 10·71-s + 2·73-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 0.417·23-s + 0.769·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s − 0.986·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s − 1.16·47-s − 49-s + 0.560·51-s − 0.274·53-s − 0.529·57-s + 0.781·59-s + 1.79·61-s − 0.481·69-s − 1.18·71-s + 0.234·73-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−16.13134491602851, −15.47538384309775, −14.77842154813901, −14.24229774991967, −13.76183514115511, −12.91548652610323, −12.66906269124138, −11.82267039910074, −11.46736354442121, −11.20351140526421, −10.26997783889035, −10.07946659107941, −9.159601842221872, −8.641610120785441, −8.071957810625793, −7.133483770927668, −6.636583943248121, −6.271887749854010, −5.424115367883541, −5.033192642126192, −4.344147691120846, −3.532237447675270, −2.820431627721373, −1.732479385667445, −0.9683890231844361, 0,
0.9683890231844361, 1.732479385667445, 2.820431627721373, 3.532237447675270, 4.344147691120846, 5.033192642126192, 5.424115367883541, 6.271887749854010, 6.636583943248121, 7.133483770927668, 8.071957810625793, 8.641610120785441, 9.159601842221872, 10.07946659107941, 10.26997783889035, 11.20351140526421, 11.46736354442121, 11.82267039910074, 12.66906269124138, 12.91548652610323, 13.76183514115511, 14.24229774991967, 14.77842154813901, 15.47538384309775, 16.13134491602851
