| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 3·11-s − 3·12-s − 13-s + 16-s + 7·17-s − 6·18-s + 19-s + 3·22-s + 4·23-s + 3·24-s + 26-s − 9·27-s + 4·29-s − 10·31-s − 32-s + 9·33-s − 7·34-s + 6·36-s − 12·37-s − 38-s + 3·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.904·11-s − 0.866·12-s − 0.277·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s + 0.229·19-s + 0.639·22-s + 0.834·23-s + 0.612·24-s + 0.196·26-s − 1.73·27-s + 0.742·29-s − 1.79·31-s − 0.176·32-s + 1.56·33-s − 1.20·34-s + 36-s − 1.97·37-s − 0.162·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.29093861191328574217466979755, −9.603959225458467302531545897476, −8.268540524730647623814161484282, −7.31089777196365830410947449224, −6.63368203908700163385624516161, −5.39555888414531273146053427777, −5.14414638880116353876067872683, −3.34911684951110812302712106923, −1.46739177433700954810449600662, 0,
1.46739177433700954810449600662, 3.34911684951110812302712106923, 5.14414638880116353876067872683, 5.39555888414531273146053427777, 6.63368203908700163385624516161, 7.31089777196365830410947449224, 8.268540524730647623814161484282, 9.603959225458467302531545897476, 10.29093861191328574217466979755
