| L(s) = 1 | − 3·3-s + 6·9-s − 3·11-s − 13-s − 17-s − 4·19-s + 4·23-s − 9·27-s + 9·29-s + 6·31-s + 9·33-s − 8·37-s + 3·39-s − 6·41-s + 8·43-s + 7·47-s + 3·51-s − 8·53-s + 12·57-s − 4·59-s + 10·61-s + 8·67-s − 12·69-s + 12·71-s − 14·73-s + 5·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.904·11-s − 0.277·13-s − 0.242·17-s − 0.917·19-s + 0.834·23-s − 1.73·27-s + 1.67·29-s + 1.07·31-s + 1.56·33-s − 1.31·37-s + 0.480·39-s − 0.937·41-s + 1.21·43-s + 1.02·47-s + 0.420·51-s − 1.09·53-s + 1.58·57-s − 0.520·59-s + 1.28·61-s + 0.977·67-s − 1.44·69-s + 1.42·71-s − 1.63·73-s + 0.562·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
| 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
−14.23286097760578, −13.57836307721289, −13.18458574595911, −12.54606264474622, −12.23248600493293, −11.91852734462797, −11.07519932345215, −10.89165121734655, −10.39879024025421, −10.01333262173060, −9.412929665485584, −8.548869492062518, −8.242089851543334, −7.444217329367860, −6.767787362388275, −6.669493575970122, −5.928224305816601, −5.405913378773341, −4.902801114024961, −4.549073638960280, −3.889169187256340, −2.909153661737918, −2.357993247562968, −1.409735320853055, −0.7101890494883200, 0,
0.7101890494883200, 1.409735320853055, 2.357993247562968, 2.909153661737918, 3.889169187256340, 4.549073638960280, 4.902801114024961, 5.405913378773341, 5.928224305816601, 6.669493575970122, 6.767787362388275, 7.444217329367860, 8.242089851543334, 8.548869492062518, 9.412929665485584, 10.01333262173060, 10.39879024025421, 10.89165121734655, 11.07519932345215, 11.91852734462797, 12.23248600493293, 12.54606264474622, 13.18458574595911, 13.57836307721289, 14.23286097760578
