| L(s) = 1 | + 2·3-s − 2·4-s − 3·7-s + 9-s − 4·12-s + 6·13-s + 4·16-s − 17-s − 6·19-s − 6·21-s + 6·23-s − 5·25-s − 4·27-s + 6·28-s − 9·29-s + 4·31-s − 2·36-s − 2·37-s + 12·39-s − 9·41-s − 6·43-s + 3·47-s + 8·48-s + 2·49-s − 2·51-s − 12·52-s − 3·53-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.13·7-s + 1/3·9-s − 1.15·12-s + 1.66·13-s + 16-s − 0.242·17-s − 1.37·19-s − 1.30·21-s + 1.25·23-s − 25-s − 0.769·27-s + 1.13·28-s − 1.67·29-s + 0.718·31-s − 1/3·36-s − 0.328·37-s + 1.92·39-s − 1.40·41-s − 0.914·43-s + 0.437·47-s + 1.15·48-s + 2/7·49-s − 0.280·51-s − 1.66·52-s − 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.614284438299390107896283197214, −8.453857897317752248942736570126, −7.31772661569686145839950009537, −6.32422609820024788903427802150, −5.63552611516237010198355246364, −4.33690474543179484375177890543, −3.60796211674639736885273377886, −3.09552236851761980264613157620, −1.70245103388878777142529973135, 0,
1.70245103388878777142529973135, 3.09552236851761980264613157620, 3.60796211674639736885273377886, 4.33690474543179484375177890543, 5.63552611516237010198355246364, 6.32422609820024788903427802150, 7.31772661569686145839950009537, 8.453857897317752248942736570126, 8.614284438299390107896283197214
