| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 13-s − 15-s + 16-s + 2·17-s − 18-s − 8·19-s + 20-s + 4·22-s − 4·23-s + 24-s + 25-s − 26-s − 27-s − 2·29-s + 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4654153823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4654153823\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−15.80052831344387, −15.30171517744854, −14.79561467624997, −14.14885403270773, −13.28098373539985, −12.99099232162931, −12.44996533027643, −11.71885893194203, −11.17325630212744, −10.59694074998635, −10.16573137570084, −9.800739144611775, −8.915604644639751, −8.345515731055043, −7.913794065501866, −7.129141203639144, −6.534354798042556, −5.944731636116802, −5.408234202387085, −4.710436474580873, −3.864821881150935, −2.996702381936694, −2.108877317836566, −1.603872832957810, −0.3167657005150278,
0.3167657005150278, 1.603872832957810, 2.108877317836566, 2.996702381936694, 3.864821881150935, 4.710436474580873, 5.408234202387085, 5.944731636116802, 6.534354798042556, 7.129141203639144, 7.913794065501866, 8.345515731055043, 8.915604644639751, 9.800739144611775, 10.16573137570084, 10.59694074998635, 11.17325630212744, 11.71885893194203, 12.44996533027643, 12.99099232162931, 13.28098373539985, 14.14885403270773, 14.79561467624997, 15.30171517744854, 15.80052831344387
