| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s − 4·10-s + 11-s − 16-s + 7·17-s − 5·19-s + 4·20-s + 22-s + 9·23-s + 11·25-s − 29-s + 2·31-s + 5·32-s + 7·34-s − 3·37-s − 5·38-s + 12·40-s + 2·41-s − 43-s − 44-s + 9·46-s − 7·47-s + 11·50-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 1.26·10-s + 0.301·11-s − 1/4·16-s + 1.69·17-s − 1.14·19-s + 0.894·20-s + 0.213·22-s + 1.87·23-s + 11/5·25-s − 0.185·29-s + 0.359·31-s + 0.883·32-s + 1.20·34-s − 0.493·37-s − 0.811·38-s + 1.89·40-s + 0.312·41-s − 0.152·43-s − 0.150·44-s + 1.32·46-s − 1.02·47-s + 1.55·50-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−7.918591191841167723658398323773, −7.23986283127684186553070220821, −6.45976208354825351371988379189, −5.50512660713362039868846413939, −4.71667334713135517179396740745, −4.24391936197214544820332942753, −3.35895861301065280817714307190, −3.04240983825500883952914853665, −1.13058554035647935294530158863, 0,
1.13058554035647935294530158863, 3.04240983825500883952914853665, 3.35895861301065280817714307190, 4.24391936197214544820332942753, 4.71667334713135517179396740745, 5.50512660713362039868846413939, 6.45976208354825351371988379189, 7.23986283127684186553070220821, 7.918591191841167723658398323773
