| L(s) = 1 | − 4·5-s − 3·7-s − 2·11-s − 6·17-s − 4·19-s + 8·23-s + 11·25-s − 4·29-s − 31-s + 12·35-s + 2·37-s + 12·41-s − 7·43-s − 6·47-s + 2·49-s + 4·53-s + 8·55-s + 61-s − 13·67-s − 12·71-s − 73-s + 6·77-s + 7·79-s − 14·83-s + 24·85-s − 10·89-s + 16·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.13·7-s − 0.603·11-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 11/5·25-s − 0.742·29-s − 0.179·31-s + 2.02·35-s + 0.328·37-s + 1.87·41-s − 1.06·43-s − 0.875·47-s + 2/7·49-s + 0.549·53-s + 1.07·55-s + 0.128·61-s − 1.58·67-s − 1.42·71-s − 0.117·73-s + 0.683·77-s + 0.787·79-s − 1.53·83-s + 2.60·85-s − 1.05·89-s + 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
| 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
−14.57164346179956, −13.52171715709120, −13.04131824869659, −12.91075758378884, −12.46946113309918, −11.69256355733422, −11.34145436203697, −10.92056991678316, −10.51620593766031, −9.852050760768407, −9.010884813778013, −8.934738166069862, −8.311040049913843, −7.659007870637598, −7.260925643350733, −6.789008025757322, −6.329121421539966, −5.613916085815534, −4.787013883586565, −4.405157474555466, −3.935284577927105, −3.163057252802732, −2.926808290290031, −2.120562497737144, −1.005365377455377, 0, 0,
1.005365377455377, 2.120562497737144, 2.926808290290031, 3.163057252802732, 3.935284577927105, 4.405157474555466, 4.787013883586565, 5.613916085815534, 6.329121421539966, 6.789008025757322, 7.260925643350733, 7.659007870637598, 8.311040049913843, 8.934738166069862, 9.010884813778013, 9.852050760768407, 10.51620593766031, 10.92056991678316, 11.34145436203697, 11.69256355733422, 12.46946113309918, 12.91075758378884, 13.04131824869659, 13.52171715709120, 14.57164346179956
