| L(s) = 1 | − 3-s − 7-s + 9-s + 2·13-s − 4·19-s + 21-s − 6·23-s − 5·25-s − 27-s + 6·29-s + 8·31-s − 2·37-s − 2·39-s − 12·41-s − 4·43-s + 12·47-s + 49-s + 6·53-s + 4·57-s − 10·61-s − 63-s − 8·67-s + 6·69-s + 6·71-s + 10·73-s + 5·75-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.917·19-s + 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.320·39-s − 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.977·67-s + 0.722·69-s + 0.712·71-s + 1.17·73-s + 0.577·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019047478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019047478\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−13.45860856419135, −12.63025114209904, −12.16993268287202, −11.97720070940539, −11.47842580583104, −10.76848105167255, −10.40380139616463, −10.05053808229669, −9.595601321017400, −8.865541226650760, −8.379389951251587, −8.087795137128879, −7.361669739739311, −6.787331919808542, −6.300543597069002, −6.047928921650866, −5.425863427490182, −4.785813551064551, −4.244977574183779, −3.813419610388346, −3.167129887210789, −2.422799894557805, −1.860757603262664, −1.124884312610490, −0.3273626574412832,
0.3273626574412832, 1.124884312610490, 1.860757603262664, 2.422799894557805, 3.167129887210789, 3.813419610388346, 4.244977574183779, 4.785813551064551, 5.425863427490182, 6.047928921650866, 6.300543597069002, 6.787331919808542, 7.361669739739311, 8.087795137128879, 8.379389951251587, 8.865541226650760, 9.595601321017400, 10.05053808229669, 10.40380139616463, 10.76848105167255, 11.47842580583104, 11.97720070940539, 12.16993268287202, 12.63025114209904, 13.45860856419135
