| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 13-s − 2·15-s + 6·17-s + 8·19-s − 8·21-s + 25-s − 4·27-s + 4·31-s + 4·35-s + 2·37-s − 2·39-s − 12·41-s + 8·43-s − 45-s − 6·47-s + 9·49-s + 12·51-s − 12·53-s + 16·57-s − 12·59-s − 8·61-s − 4·63-s + 65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 1.45·17-s + 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 0.718·31-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 1.87·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.68·51-s − 1.64·53-s + 2.11·57-s − 1.56·59-s − 1.02·61-s − 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.85402356804568, −13.42985445291115, −12.75573224432769, −12.42296634912733, −11.88796704312467, −11.56415572119363, −10.70299354963975, −10.20764972427822, −9.604591637795067, −9.469062060312869, −9.073336294263561, −8.189643456930420, −7.917395042957876, −7.480091072625763, −6.971420961455906, −6.260090020640049, −5.885701766527669, −5.108678308352581, −4.641114884185999, −3.600175463603405, −3.381126443294354, −3.118127352761736, −2.531204167447898, −1.607121197567569, −0.8681432636641450, 0,
0.8681432636641450, 1.607121197567569, 2.531204167447898, 3.118127352761736, 3.381126443294354, 3.600175463603405, 4.641114884185999, 5.108678308352581, 5.885701766527669, 6.260090020640049, 6.971420961455906, 7.480091072625763, 7.917395042957876, 8.189643456930420, 9.073336294263561, 9.469062060312869, 9.604591637795067, 10.20764972427822, 10.70299354963975, 11.56415572119363, 11.88796704312467, 12.42296634912733, 12.75573224432769, 13.42985445291115, 13.85402356804568
