| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 16-s − 2·17-s + 2·19-s − 20-s + 22-s − 6·23-s + 25-s − 2·29-s − 32-s + 2·34-s − 6·37-s − 2·38-s + 40-s − 2·41-s − 2·43-s − 44-s + 6·46-s + 2·47-s − 7·49-s − 50-s − 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 1/4·16-s − 0.485·17-s + 0.458·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.371·29-s − 0.176·32-s + 0.342·34-s − 0.986·37-s − 0.324·38-s + 0.158·40-s − 0.312·41-s − 0.304·43-s − 0.150·44-s + 0.884·46-s + 0.291·47-s − 49-s − 0.141·50-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−9.585061788865674397535829002795, −8.681551793214488744099226686983, −7.980074178056792123593772000541, −7.22701879062173689410571032674, −6.32391276808398118475394591840, −5.29499086428259716447525409361, −4.12386584858564076185898674350, −3.01181279495610934783547817167, −1.72432388001961117188035940243, 0,
1.72432388001961117188035940243, 3.01181279495610934783547817167, 4.12386584858564076185898674350, 5.29499086428259716447525409361, 6.32391276808398118475394591840, 7.22701879062173689410571032674, 7.980074178056792123593772000541, 8.681551793214488744099226686983, 9.585061788865674397535829002795
