| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 6·11-s + 12-s − 13-s + 16-s − 2·17-s + 18-s + 2·19-s + 6·22-s + 4·23-s + 24-s − 26-s + 27-s + 8·29-s − 4·31-s + 32-s + 6·33-s − 2·34-s + 36-s − 2·37-s + 2·38-s − 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s + 1.27·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.176·32-s + 1.04·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.730202452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.730202452\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−13.83709789342754, −13.32908112771882, −13.01921785316343, −12.28665471309472, −11.81195912270793, −11.61805921141342, −11.00616643421082, −10.18108645684983, −9.979399444548530, −9.265416019076131, −8.675970224464870, −8.542967388023659, −7.642285254337306, −7.078498960788414, −6.631130868317651, −6.386527076738203, −5.505707993254293, −4.834998428485891, −4.561040610769045, −3.690824069852586, −3.444686350936847, −2.820696078786231, −1.954092474845064, −1.514802040500580, −0.7064836886121870,
0.7064836886121870, 1.514802040500580, 1.954092474845064, 2.820696078786231, 3.444686350936847, 3.690824069852586, 4.561040610769045, 4.834998428485891, 5.505707993254293, 6.386527076738203, 6.631130868317651, 7.078498960788414, 7.642285254337306, 8.542967388023659, 8.675970224464870, 9.265416019076131, 9.979399444548530, 10.18108645684983, 11.00616643421082, 11.61805921141342, 11.81195912270793, 12.28665471309472, 13.01921785316343, 13.32908112771882, 13.83709789342754
