| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 2·11-s − 12-s + 4·13-s − 16-s + 6·17-s − 18-s − 6·19-s + 2·22-s − 8·23-s + 3·24-s − 4·26-s + 27-s − 29-s + 6·31-s − 5·32-s − 2·33-s − 6·34-s − 36-s + 6·37-s + 6·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 1.10·13-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.37·19-s + 0.426·22-s − 1.66·23-s + 0.612·24-s − 0.784·26-s + 0.192·27-s − 0.185·29-s + 1.07·31-s − 0.883·32-s − 0.348·33-s − 1.02·34-s − 1/6·36-s + 0.986·37-s + 0.973·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132672123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132672123\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.80299946895638, −13.13744222399561, −12.89521313259147, −12.29809050601132, −11.76799942687439, −11.00207335700061, −10.53066965761414, −10.26858980261322, −9.641809998730644, −9.304595948315018, −8.648150881341676, −8.204348390787603, −7.880478770059216, −7.594073078831347, −6.680236499684444, −6.067238443649349, −5.694349244057800, −4.893983258997706, −4.272539156030521, −3.866142379914932, −3.333011281627458, −2.379866537937491, −1.995451228843714, −1.034062076344056, −0.5993226966858009,
0.5993226966858009, 1.034062076344056, 1.995451228843714, 2.379866537937491, 3.333011281627458, 3.866142379914932, 4.272539156030521, 4.893983258997706, 5.694349244057800, 6.067238443649349, 6.680236499684444, 7.594073078831347, 7.880478770059216, 8.204348390787603, 8.648150881341676, 9.304595948315018, 9.641809998730644, 10.26858980261322, 10.53066965761414, 11.00207335700061, 11.76799942687439, 12.29809050601132, 12.89521313259147, 13.13744222399561, 13.80299946895638
