| L(s) = 1 | + 3-s − 4·7-s − 2·9-s + 2·11-s − 5·13-s + 17-s − 7·19-s − 4·21-s − 8·23-s − 5·27-s − 5·29-s − 3·31-s + 2·33-s + 4·37-s − 5·39-s − 4·41-s + 3·47-s + 9·49-s + 51-s − 7·53-s − 7·57-s − 3·59-s − 11·61-s + 8·63-s + 10·67-s − 8·69-s + 71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s + 0.603·11-s − 1.38·13-s + 0.242·17-s − 1.60·19-s − 0.872·21-s − 1.66·23-s − 0.962·27-s − 0.928·29-s − 0.538·31-s + 0.348·33-s + 0.657·37-s − 0.800·39-s − 0.624·41-s + 0.437·47-s + 9/7·49-s + 0.140·51-s − 0.961·53-s − 0.927·57-s − 0.390·59-s − 1.40·61-s + 1.00·63-s + 1.22·67-s − 0.963·69-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−15.67344560706192, −15.17007637178085, −14.74461792346743, −14.14864586421441, −13.85039065883447, −12.89981438324642, −12.79124359763769, −12.11477128039735, −11.65312550814975, −10.86106051516403, −10.23570884386607, −9.682183942643890, −9.345965715347775, −8.800718890937527, −8.088131832664943, −7.575538310292148, −6.881751508646886, −6.222775025358365, −5.933355871780180, −5.065493115453207, −4.100191517562934, −3.775690677630288, −2.938337426830058, −2.429112637788205, −1.729302770763276, 0, 0,
1.729302770763276, 2.429112637788205, 2.938337426830058, 3.775690677630288, 4.100191517562934, 5.065493115453207, 5.933355871780180, 6.222775025358365, 6.881751508646886, 7.575538310292148, 8.088131832664943, 8.800718890937527, 9.345965715347775, 9.682183942643890, 10.23570884386607, 10.86106051516403, 11.65312550814975, 12.11477128039735, 12.79124359763769, 12.89981438324642, 13.85039065883447, 14.14864586421441, 14.74461792346743, 15.17007637178085, 15.67344560706192
