| L(s) = 1 | − 3-s + 7-s + 9-s + 6·13-s − 17-s − 2·19-s − 21-s + 4·23-s − 27-s + 4·31-s − 10·37-s − 6·39-s − 2·41-s − 8·47-s + 49-s + 51-s + 6·53-s + 2·57-s + 4·59-s − 8·61-s + 63-s + 4·67-s − 4·69-s − 8·71-s − 8·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.242·17-s − 0.458·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.718·31-s − 1.64·37-s − 0.960·39-s − 0.312·41-s − 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s − 0.949·71-s − 0.900·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| 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
−15.10152904352112, −14.88978150631845, −13.93051820848763, −13.64930679708050, −13.10385616571868, −12.58425702525438, −11.92628280175993, −11.44902954782999, −10.99594072937199, −10.51688498793313, −10.06629636870951, −9.221894481998283, −8.612488791275272, −8.401908825337755, −7.551522462408015, −6.911857584854875, −6.387398001069845, −5.927131284297871, −5.159616928018445, −4.741714110391973, −3.909864336991818, −3.453718106989578, −2.546697232973249, −1.596961672196941, −1.104276469749406, 0,
1.104276469749406, 1.596961672196941, 2.546697232973249, 3.453718106989578, 3.909864336991818, 4.741714110391973, 5.159616928018445, 5.927131284297871, 6.387398001069845, 6.911857584854875, 7.551522462408015, 8.401908825337755, 8.612488791275272, 9.221894481998283, 10.06629636870951, 10.51688498793313, 10.99594072937199, 11.44902954782999, 11.92628280175993, 12.58425702525438, 13.10385616571868, 13.64930679708050, 13.93051820848763, 14.88978150631845, 15.10152904352112
