| L(s) = 1 | − 2·5-s + 4·7-s + 6·13-s + 17-s + 2·19-s + 4·23-s − 25-s − 4·29-s − 2·31-s − 8·35-s + 9·37-s − 5·41-s − 4·43-s − 6·47-s + 9·49-s − 9·53-s + 11·59-s + 11·61-s − 12·65-s − 2·67-s + 3·71-s − 12·73-s + 8·79-s + 11·83-s − 2·85-s + 2·89-s + 24·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.66·13-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 1/5·25-s − 0.742·29-s − 0.359·31-s − 1.35·35-s + 1.47·37-s − 0.780·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s − 1.23·53-s + 1.43·59-s + 1.40·61-s − 1.48·65-s − 0.244·67-s + 0.356·71-s − 1.40·73-s + 0.900·79-s + 1.20·83-s − 0.216·85-s + 0.211·89-s + 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
| 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
−13.08545836703341, −12.37610019462588, −11.73964603059980, −11.53573746575649, −11.20390471637043, −10.85298864324769, −10.34941869171385, −9.629173039527894, −9.178067154835182, −8.626463069969612, −8.214677058144415, −7.811799048997322, −7.662414760152792, −6.742762096008492, −6.546040071505351, −5.677178799115172, −5.305614876064287, −4.892429402573875, −4.152694989099792, −3.877139347777303, −3.373168171267687, −2.704428256188176, −1.904842510017521, −1.339833827605425, −0.9663133378199315, 0,
0.9663133378199315, 1.339833827605425, 1.904842510017521, 2.704428256188176, 3.373168171267687, 3.877139347777303, 4.152694989099792, 4.892429402573875, 5.305614876064287, 5.677178799115172, 6.546040071505351, 6.742762096008492, 7.662414760152792, 7.811799048997322, 8.214677058144415, 8.626463069969612, 9.178067154835182, 9.629173039527894, 10.34941869171385, 10.85298864324769, 11.20390471637043, 11.53573746575649, 11.73964603059980, 12.37610019462588, 13.08545836703341
