| L(s) = 1 | + 2·3-s + 7-s + 9-s + 11-s − 2·13-s + 17-s + 5·19-s + 2·21-s + 6·23-s − 4·27-s − 5·29-s + 7·31-s + 2·33-s − 6·37-s − 4·39-s + 9·41-s − 2·43-s + 49-s + 2·51-s + 10·57-s − 7·59-s − 4·61-s + 63-s − 4·67-s + 12·69-s − 8·71-s + 11·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.242·17-s + 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.769·27-s − 0.928·29-s + 1.25·31-s + 0.348·33-s − 0.986·37-s − 0.640·39-s + 1.40·41-s − 0.304·43-s + 1/7·49-s + 0.280·51-s + 1.32·57-s − 0.911·59-s − 0.512·61-s + 0.125·63-s − 0.488·67-s + 1.44·69-s − 0.949·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
| 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
−13.50602136268886, −12.97159920630489, −12.41622572761709, −11.92289458334726, −11.58963810899712, −10.84321038254774, −10.65929213557348, −9.772959251605313, −9.406276079260386, −9.230927813835523, −8.538326020128537, −8.150160773540919, −7.564953695414799, −7.352606996261341, −6.727688533312424, −6.056048512558218, −5.466256611868456, −4.950255463817775, −4.493197912774457, −3.651593206373218, −3.416182980152378, −2.647443652422740, −2.421153979399761, −1.480007327572590, −1.076326709808020, 0,
1.076326709808020, 1.480007327572590, 2.421153979399761, 2.647443652422740, 3.416182980152378, 3.651593206373218, 4.493197912774457, 4.950255463817775, 5.466256611868456, 6.056048512558218, 6.727688533312424, 7.352606996261341, 7.564953695414799, 8.150160773540919, 8.538326020128537, 9.230927813835523, 9.406276079260386, 9.772959251605313, 10.65929213557348, 10.84321038254774, 11.58963810899712, 11.92289458334726, 12.41622572761709, 12.97159920630489, 13.50602136268886
