| L(s) = 1 | + 3·3-s + 6·9-s − 11-s − 4·13-s + 5·17-s + 19-s + 2·23-s + 9·27-s + 8·29-s − 10·31-s − 3·33-s − 6·37-s − 12·39-s + 3·41-s + 4·43-s + 4·47-s + 15·51-s + 6·53-s + 3·57-s + 8·59-s + 10·61-s − 67-s + 6·69-s − 12·71-s + 3·73-s + 6·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s − 0.301·11-s − 1.10·13-s + 1.21·17-s + 0.229·19-s + 0.417·23-s + 1.73·27-s + 1.48·29-s − 1.79·31-s − 0.522·33-s − 0.986·37-s − 1.92·39-s + 0.468·41-s + 0.609·43-s + 0.583·47-s + 2.10·51-s + 0.824·53-s + 0.397·57-s + 1.04·59-s + 1.28·61-s − 0.122·67-s + 0.722·69-s − 1.42·71-s + 0.351·73-s + 0.675·79-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
| \(L(1)\) |
\(\approx\) |
\(5.407680074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.407680074\) |
| \(L(\frac{3}{2})\) |
|
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.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−14.21520763856716, −13.56568832278227, −13.17984202348434, −12.54751642730167, −12.24284667029749, −11.70883028409900, −10.78069935062193, −10.34770385251737, −9.908386116164001, −9.396444246435909, −8.956305326520192, −8.482677668029123, −7.866998690892141, −7.490462915806019, −7.120859283835680, −6.489813741180604, −5.385283639285039, −5.261376211508720, −4.344847661149915, −3.792506968307078, −3.268017292575485, −2.655855881054588, −2.269357264021972, −1.490957144088998, −0.6775058644739289,
0.6775058644739289, 1.490957144088998, 2.269357264021972, 2.655855881054588, 3.268017292575485, 3.792506968307078, 4.344847661149915, 5.261376211508720, 5.385283639285039, 6.489813741180604, 7.120859283835680, 7.490462915806019, 7.866998690892141, 8.482677668029123, 8.956305326520192, 9.396444246435909, 9.908386116164001, 10.34770385251737, 10.78069935062193, 11.70883028409900, 12.24284667029749, 12.54751642730167, 13.17984202348434, 13.56568832278227, 14.21520763856716
