| L(s) = 1 | + 2·5-s + 4·11-s + 2·13-s − 2·17-s − 23-s − 25-s + 2·29-s − 10·37-s − 6·41-s + 8·43-s − 8·47-s + 6·53-s + 8·55-s − 4·59-s − 14·61-s + 4·65-s + 8·67-s + 8·71-s + 6·73-s + 12·79-s − 12·83-s − 4·85-s − 2·89-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.208·23-s − 1/5·25-s + 0.371·29-s − 1.64·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 0.824·53-s + 1.07·55-s − 0.520·59-s − 1.79·61-s + 0.496·65-s + 0.977·67-s + 0.949·71-s + 0.702·73-s + 1.35·79-s − 1.31·83-s − 0.433·85-s − 0.211·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.02860349891953, −13.80249067460181, −13.44941549403366, −12.66221774152881, −12.31481011584845, −11.75178948501478, −11.24013913605201, −10.71865061290503, −10.20986531510378, −9.655290557156642, −9.230614274877544, −8.760323083131578, −8.293332755909418, −7.587710751751084, −6.901993447626375, −6.396993823684295, −6.184572693359030, −5.364687645579010, −4.974674087712420, −4.099108096437878, −3.740883732022805, −3.013329766460846, −2.206277035405580, −1.655663339827161, −1.108906820459594, 0,
1.108906820459594, 1.655663339827161, 2.206277035405580, 3.013329766460846, 3.740883732022805, 4.099108096437878, 4.974674087712420, 5.364687645579010, 6.184572693359030, 6.396993823684295, 6.901993447626375, 7.587710751751084, 8.293332755909418, 8.760323083131578, 9.230614274877544, 9.655290557156642, 10.20986531510378, 10.71865061290503, 11.24013913605201, 11.75178948501478, 12.31481011584845, 12.66221774152881, 13.44941549403366, 13.80249067460181, 14.02860349891953
