| L(s) = 1 | + 3-s + 5-s − 5·7-s − 2·9-s + 2·11-s + 15-s − 3·17-s + 2·19-s − 5·21-s + 4·23-s − 4·25-s − 5·27-s − 6·29-s + 4·31-s + 2·33-s − 5·35-s − 11·37-s − 8·41-s − 43-s − 2·45-s − 9·47-s + 18·49-s − 3·51-s − 12·53-s + 2·55-s + 2·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s − 2/3·9-s + 0.603·11-s + 0.258·15-s − 0.727·17-s + 0.458·19-s − 1.09·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.845·35-s − 1.80·37-s − 1.24·41-s − 0.152·43-s − 0.298·45-s − 1.31·47-s + 18/7·49-s − 0.420·51-s − 1.64·53-s + 0.269·55-s + 0.264·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−9.256709491990158107296756353137, −8.687847923613485910367058066152, −7.52572312819139412231097737039, −6.56613787924881641502302975467, −6.13975226466306178906887942213, −5.04019594289518523427290782366, −3.54343108600467791289032395101, −3.18916207013444867550122856798, −1.96354020987093639836026474240, 0,
1.96354020987093639836026474240, 3.18916207013444867550122856798, 3.54343108600467791289032395101, 5.04019594289518523427290782366, 6.13975226466306178906887942213, 6.56613787924881641502302975467, 7.52572312819139412231097737039, 8.687847923613485910367058066152, 9.256709491990158107296756353137
