| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 2·13-s − 2·15-s − 4·17-s − 8·21-s + 25-s − 4·27-s − 29-s + 4·35-s − 4·37-s − 4·39-s − 2·41-s − 2·43-s − 45-s − 10·47-s + 9·49-s − 8·51-s − 6·53-s + 4·59-s − 6·61-s − 4·63-s + 2·65-s + 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.74·21-s + 1/5·25-s − 0.769·27-s − 0.185·29-s + 0.676·35-s − 0.657·37-s − 0.640·39-s − 0.312·41-s − 0.304·43-s − 0.149·45-s − 1.45·47-s + 9/7·49-s − 1.12·51-s − 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.503·63-s + 0.248·65-s + 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−9.349740516704699203363844986193, −8.644696002225381394716955427633, −7.83558088535920311094416934297, −6.94567167793673253004427058387, −6.25906525122289254500513571613, −4.92237265927611473540229334834, −3.73203554398500870130107084153, −3.14241883887777569262642920130, −2.17765110356406377414745807873, 0,
2.17765110356406377414745807873, 3.14241883887777569262642920130, 3.73203554398500870130107084153, 4.92237265927611473540229334834, 6.25906525122289254500513571613, 6.94567167793673253004427058387, 7.83558088535920311094416934297, 8.644696002225381394716955427633, 9.349740516704699203363844986193
