| L(s) = 1 | − 2·3-s − 7-s + 9-s − 4·13-s − 4·19-s + 2·21-s + 6·23-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s + 8·39-s + 6·41-s + 4·43-s + 6·47-s + 49-s + 6·53-s + 8·57-s + 10·61-s − 63-s + 2·67-s − 12·69-s − 12·71-s + 8·73-s − 16·79-s − 11·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s + 0.436·21-s + 1.25·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s + 1.05·57-s + 1.28·61-s − 0.125·63-s + 0.244·67-s − 1.44·69-s − 1.42·71-s + 0.936·73-s − 1.80·79-s − 1.22·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82155946106456, −12.22266837354382, −12.00313759651193, −11.41884090469048, −11.02802362867919, −10.58174764206033, −10.12987660191659, −9.842308197578019, −9.141342213465615, −8.682057212023689, −8.348660350799473, −7.530342379234830, −7.121924908339090, −6.776661510418154, −6.226919591104289, −5.810759301605550, −5.341168965891515, −4.751746641846510, −4.498422178198947, −3.877889615849069, −3.045616798745966, −2.624125912413944, −2.149697425716917, −1.097815217899884, −0.7096046499702719, 0,
0.7096046499702719, 1.097815217899884, 2.149697425716917, 2.624125912413944, 3.045616798745966, 3.877889615849069, 4.498422178198947, 4.751746641846510, 5.341168965891515, 5.810759301605550, 6.226919591104289, 6.776661510418154, 7.121924908339090, 7.530342379234830, 8.348660350799473, 8.682057212023689, 9.141342213465615, 9.842308197578019, 10.12987660191659, 10.58174764206033, 11.02802362867919, 11.41884090469048, 12.00313759651193, 12.22266837354382, 12.82155946106456
