| L(s) = 1 | + 5-s − 11-s + 2·13-s + 3·17-s + 5·19-s + 3·23-s − 4·25-s − 6·29-s + 31-s + 5·37-s − 10·41-s + 4·43-s + 47-s − 9·53-s − 55-s − 3·59-s + 3·61-s + 2·65-s − 11·67-s − 16·71-s − 7·73-s − 11·79-s + 4·83-s + 3·85-s − 9·89-s + 5·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.727·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s − 1.11·29-s + 0.179·31-s + 0.821·37-s − 1.56·41-s + 0.609·43-s + 0.145·47-s − 1.23·53-s − 0.134·55-s − 0.390·59-s + 0.384·61-s + 0.248·65-s − 1.34·67-s − 1.89·71-s − 0.819·73-s − 1.23·79-s + 0.439·83-s + 0.325·85-s − 0.953·89-s + 0.512·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 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.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 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.ab |
| 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.l |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−15.53146241310813, −14.92103918428167, −14.38096442364256, −13.87233955728400, −13.24527655602769, −13.07675303834670, −12.23461704862250, −11.65561557303952, −11.31560810756328, −10.50973098810586, −10.11823966008227, −9.457136021804975, −9.100383366021116, −8.336648185247202, −7.686651282359549, −7.331207770429055, −6.533462691284026, −5.771640917102375, −5.563909166296976, −4.764986675602535, −4.028796389096090, −3.243660680264090, −2.800309342271864, −1.717871400612755, −1.213863507726078, 0,
1.213863507726078, 1.717871400612755, 2.800309342271864, 3.243660680264090, 4.028796389096090, 4.764986675602535, 5.563909166296976, 5.771640917102375, 6.533462691284026, 7.331207770429055, 7.686651282359549, 8.336648185247202, 9.100383366021116, 9.457136021804975, 10.11823966008227, 10.50973098810586, 11.31560810756328, 11.65561557303952, 12.23461704862250, 13.07675303834670, 13.24527655602769, 13.87233955728400, 14.38096442364256, 14.92103918428167, 15.53146241310813
