| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 6·11-s + 2·13-s − 4·14-s + 16-s − 4·19-s − 6·22-s − 5·25-s − 2·26-s + 4·28-s + 4·31-s − 32-s + 4·37-s + 4·38-s + 6·41-s + 8·43-s + 6·44-s + 9·49-s + 5·50-s + 2·52-s + 6·53-s − 4·56-s + 4·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 1.80·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s − 1.27·22-s − 25-s − 0.392·26-s + 0.755·28-s + 0.718·31-s − 0.176·32-s + 0.657·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.904·44-s + 9/7·49-s + 0.707·50-s + 0.277·52-s + 0.824·53-s − 0.534·56-s + 0.512·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088293764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088293764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.243514991730539144217181571602, −7.70798233132200076449980309283, −6.82004608046967562037073412090, −6.19743537938870376645485282745, −5.44433194423880119523961633110, −4.21154689020545323480297501674, −4.02792188414833404462281994143, −2.52693971063750362367950493388, −1.64644018076655648104085466620, −0.968095230515067382105649946659,
0.968095230515067382105649946659, 1.64644018076655648104085466620, 2.52693971063750362367950493388, 4.02792188414833404462281994143, 4.21154689020545323480297501674, 5.44433194423880119523961633110, 6.19743537938870376645485282745, 6.82004608046967562037073412090, 7.70798233132200076449980309283, 8.243514991730539144217181571602
