| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s − 4·11-s − 2·12-s − 2·13-s − 15-s − 4·16-s − 4·17-s + 2·18-s + 4·19-s + 2·20-s − 8·22-s − 6·23-s + 25-s − 4·26-s − 27-s + 4·29-s − 2·30-s − 6·31-s − 8·32-s + 4·33-s − 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 0.258·15-s − 16-s − 0.970·17-s + 0.471·18-s + 0.917·19-s + 0.447·20-s − 1.70·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.365·30-s − 1.07·31-s − 1.41·32-s + 0.696·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 163 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−8.481908347190688674159689893256, −7.49183026862088891654908031750, −6.71420026382494853880259729802, −5.96405696957633305125210433609, −5.24727898748485773824017415428, −4.82864864919859547229153842865, −3.84935171716394412914703021741, −2.82985322021221654045624734472, −1.98188871732766678776780397352, 0,
1.98188871732766678776780397352, 2.82985322021221654045624734472, 3.84935171716394412914703021741, 4.82864864919859547229153842865, 5.24727898748485773824017415428, 5.96405696957633305125210433609, 6.71420026382494853880259729802, 7.49183026862088891654908031750, 8.481908347190688674159689893256
