| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 4·7-s + 8-s + 9-s + 10-s − 2·12-s − 13-s + 4·14-s − 2·15-s + 16-s + 6·17-s + 18-s − 2·19-s + 20-s − 8·21-s + 6·23-s − 2·24-s + 25-s − 26-s + 4·27-s + 4·28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s − 0.277·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 1.74·21-s + 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.196·26-s + 0.769·27-s + 0.755·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.618203060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.618203060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.20003189714115, −15.23092136347794, −14.75042657419212, −14.42429354988992, −13.85403652022262, −13.15423587858689, −12.50087830900113, −12.03151180040867, −11.55557893410011, −11.07110499087718, −10.55242021684803, −10.05233423180403, −9.205579517387938, −8.268439646493397, −7.957238021447144, −7.019384646528709, −6.558272346303689, −5.703370121483604, −5.383561253733787, −4.794743156872237, −4.370519429162164, −3.221355782380673, −2.483180293523739, −1.463674587309159, −0.8708900478600117,
0.8708900478600117, 1.463674587309159, 2.483180293523739, 3.221355782380673, 4.370519429162164, 4.794743156872237, 5.383561253733787, 5.703370121483604, 6.558272346303689, 7.019384646528709, 7.957238021447144, 8.268439646493397, 9.205579517387938, 10.05233423180403, 10.55242021684803, 11.07110499087718, 11.55557893410011, 12.03151180040867, 12.50087830900113, 13.15423587858689, 13.85403652022262, 14.42429354988992, 14.75042657419212, 15.23092136347794, 16.20003189714115
