| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s − 2·9-s − 10-s − 2·11-s + 12-s + 5·13-s − 15-s + 16-s + 17-s − 2·18-s + 3·19-s − 20-s − 2·22-s − 2·23-s + 24-s + 25-s + 5·26-s − 5·27-s − 5·29-s − 30-s + 3·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.38·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.962·27-s − 0.928·29-s − 0.182·30-s + 0.538·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.628398445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.628398445\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75224010783973914672660495951, −7.24078121909324223452972252976, −6.22014694115145208675356613037, −5.72285015949319826565276029030, −5.05797657522451710283424979289, −4.00930814085351732037120151496, −3.57203019074346732517166482547, −2.84561943520071171619936994705, −2.04669458053518419971780046608, −0.811387849095929839202988539350,
0.811387849095929839202988539350, 2.04669458053518419971780046608, 2.84561943520071171619936994705, 3.57203019074346732517166482547, 4.00930814085351732037120151496, 5.05797657522451710283424979289, 5.72285015949319826565276029030, 6.22014694115145208675356613037, 7.24078121909324223452972252976, 7.75224010783973914672660495951
