| L(s) = 1 | − 3-s − 2·5-s + 9-s + 13-s + 2·15-s + 2·17-s + 6·19-s + 8·23-s − 25-s − 27-s + 4·29-s + 4·31-s + 6·37-s − 39-s − 8·43-s − 2·45-s − 2·51-s − 6·57-s + 8·59-s + 10·61-s − 2·65-s − 14·67-s − 8·69-s + 8·71-s + 2·73-s + 75-s − 14·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.277·13-s + 0.516·15-s + 0.485·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.986·37-s − 0.160·39-s − 1.21·43-s − 0.298·45-s − 0.280·51-s − 0.794·57-s + 1.04·59-s + 1.28·61-s − 0.248·65-s − 1.71·67-s − 0.963·69-s + 0.949·71-s + 0.234·73-s + 0.115·75-s − 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824813514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824813514\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−12.93339641317492, −12.24818406039849, −11.79416978217478, −11.59122306195300, −11.18861269027482, −10.64579111855606, −9.994889334745492, −9.842807693914465, −9.078612658351469, −8.688191875751036, −8.018483322529016, −7.729623062971465, −7.209806742680057, −6.682988122210186, −6.330295327575645, −5.510988845533735, −5.196724874741468, −4.760383847235934, −3.999530382075832, −3.731343746296796, −2.878910649825344, −2.720043194147084, −1.463313112659725, −1.091730079559299, −0.4529627045689718,
0.4529627045689718, 1.091730079559299, 1.463313112659725, 2.720043194147084, 2.878910649825344, 3.731343746296796, 3.999530382075832, 4.760383847235934, 5.196724874741468, 5.510988845533735, 6.330295327575645, 6.682988122210186, 7.209806742680057, 7.729623062971465, 8.018483322529016, 8.688191875751036, 9.078612658351469, 9.842807693914465, 9.994889334745492, 10.64579111855606, 11.18861269027482, 11.59122306195300, 11.79416978217478, 12.24818406039849, 12.93339641317492
