| L(s) = 1 | + 3·3-s + 2·5-s − 4·7-s + 6·9-s − 2·11-s + 5·13-s + 6·15-s + 4·17-s + 2·19-s − 12·21-s + 23-s − 25-s + 9·27-s + 7·29-s − 3·31-s − 6·33-s − 8·35-s − 2·37-s + 15·39-s − 9·41-s + 8·43-s + 12·45-s + 9·47-s + 9·49-s + 12·51-s − 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 1.51·7-s + 2·9-s − 0.603·11-s + 1.38·13-s + 1.54·15-s + 0.970·17-s + 0.458·19-s − 2.61·21-s + 0.208·23-s − 1/5·25-s + 1.73·27-s + 1.29·29-s − 0.538·31-s − 1.04·33-s − 1.35·35-s − 0.328·37-s + 2.40·39-s − 1.40·41-s + 1.21·43-s + 1.78·45-s + 1.31·47-s + 9/7·49-s + 1.68·51-s − 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.328413675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.328413675\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−9.329484681401310670308265163855, −8.902069120921913079823685276690, −8.046180103929811076438808036308, −7.20276326639128304407472806256, −6.30739888458901592854412022981, −5.52266446656413560027397663397, −3.99096260128289979003484172303, −3.21402252057952047529474757459, −2.65858804857211710860724509971, −1.38511650780934413670188743730,
1.38511650780934413670188743730, 2.65858804857211710860724509971, 3.21402252057952047529474757459, 3.99096260128289979003484172303, 5.52266446656413560027397663397, 6.30739888458901592854412022981, 7.20276326639128304407472806256, 8.046180103929811076438808036308, 8.902069120921913079823685276690, 9.329484681401310670308265163855
