| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 2·13-s − 3·15-s + 3·17-s + 5·19-s − 3·23-s − 4·25-s − 9·27-s + 6·29-s + 31-s − 5·37-s − 6·39-s − 10·41-s + 4·43-s + 6·45-s − 47-s − 9·51-s − 9·53-s − 15·57-s − 3·59-s + 3·61-s + 2·65-s + 11·67-s + 9·69-s + 16·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.554·13-s − 0.774·15-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 4/5·25-s − 1.73·27-s + 1.11·29-s + 0.179·31-s − 0.821·37-s − 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.894·45-s − 0.145·47-s − 1.26·51-s − 1.23·53-s − 1.98·57-s − 0.390·59-s + 0.384·61-s + 0.248·65-s + 1.34·67-s + 1.08·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.96522844055372, −14.04703009606147, −13.83738356457826, −13.32357995335630, −12.46228573260822, −12.20815893558794, −11.86120467650913, −11.13172636559390, −10.88235115120957, −10.16303813396961, −9.787026252252774, −9.433291940477367, −8.369273567575970, −7.972875805402506, −7.218616389834685, −6.586450866580328, −6.283776394984190, −5.624428851580821, −5.191561177529108, −4.807881107482157, −3.855550985300810, −3.407990012061900, −2.320805705078825, −1.448471791250634, −0.9493308912318411, 0,
0.9493308912318411, 1.448471791250634, 2.320805705078825, 3.407990012061900, 3.855550985300810, 4.807881107482157, 5.191561177529108, 5.624428851580821, 6.283776394984190, 6.586450866580328, 7.218616389834685, 7.972875805402506, 8.369273567575970, 9.433291940477367, 9.787026252252774, 10.16303813396961, 10.88235115120957, 11.13172636559390, 11.86120467650913, 12.20815893558794, 12.46228573260822, 13.32357995335630, 13.83738356457826, 14.04703009606147, 14.96522844055372
