| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 4·11-s + 2·13-s − 4·14-s + 16-s + 2·19-s + 4·22-s − 4·23-s − 5·25-s − 2·26-s + 4·28-s − 2·31-s − 32-s − 10·37-s − 2·38-s − 12·41-s + 10·43-s − 4·44-s + 4·46-s − 47-s + 9·49-s + 5·50-s + 2·52-s − 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.458·19-s + 0.852·22-s − 0.834·23-s − 25-s − 0.392·26-s + 0.755·28-s − 0.359·31-s − 0.176·32-s − 1.64·37-s − 0.324·38-s − 1.87·41-s + 1.52·43-s − 0.603·44-s + 0.589·46-s − 0.145·47-s + 9/7·49-s + 0.707·50-s + 0.277·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6473879866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6473879866\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| 47 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−12.78819758432932, −12.24515987812758, −11.85670385448605, −11.39378384544755, −10.91897433540166, −10.66157865822176, −10.07986153321155, −9.712055466031855, −9.082406851335472, −8.388409335000826, −8.283948245980558, −7.872242123016391, −7.309671687382505, −6.947871165869789, −6.167848947964678, −5.566511886517923, −5.315025666065385, −4.765073708800431, −4.052239001063258, −3.582157381468144, −2.798172464376001, −2.268958637282775, −1.564067739002409, −1.397155318268471, −0.2375444270841942,
0.2375444270841942, 1.397155318268471, 1.564067739002409, 2.268958637282775, 2.798172464376001, 3.582157381468144, 4.052239001063258, 4.765073708800431, 5.315025666065385, 5.566511886517923, 6.167848947964678, 6.947871165869789, 7.309671687382505, 7.872242123016391, 8.283948245980558, 8.388409335000826, 9.082406851335472, 9.712055466031855, 10.07986153321155, 10.66157865822176, 10.91897433540166, 11.39378384544755, 11.85670385448605, 12.24515987812758, 12.78819758432932
