| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 3·8-s + 9-s − 10-s + 2·12-s + 2·13-s − 2·15-s − 16-s − 18-s − 6·19-s − 20-s + 8·23-s − 6·24-s + 25-s − 2·26-s + 4·27-s − 29-s + 2·30-s − 2·31-s − 5·32-s − 36-s − 2·37-s + 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.554·13-s − 0.516·15-s − 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 1.66·23-s − 1.22·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.185·29-s + 0.365·30-s − 0.359·31-s − 0.883·32-s − 1/6·36-s − 0.328·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−7.63680977417626319860156395521, −6.72958045417630691314647175969, −6.30624283276288604819269889607, −5.39830270417944638364994832918, −4.93637363698639905450550378985, −4.18882709790670868998904430555, −3.14955667191754086863496658646, −1.87520793646228447876469158892, −0.981480171235709942798260770225, 0,
0.981480171235709942798260770225, 1.87520793646228447876469158892, 3.14955667191754086863496658646, 4.18882709790670868998904430555, 4.93637363698639905450550378985, 5.39830270417944638364994832918, 6.30624283276288604819269889607, 6.72958045417630691314647175969, 7.63680977417626319860156395521
