| L(s) = 1 | − 2·5-s − 3·9-s − 6·13-s − 2·17-s − 25-s + 10·29-s − 2·37-s + 10·41-s + 6·45-s − 7·49-s + 14·53-s − 10·61-s + 12·65-s − 6·73-s + 9·81-s + 4·85-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.66·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 0.328·37-s + 1.56·41-s + 0.894·45-s − 49-s + 1.92·53-s − 1.28·61-s + 1.48·65-s − 0.702·73-s + 81-s + 0.433·85-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−15.44978779064177, −14.60531133268716, −14.35829482704737, −13.78972954427078, −13.13413735623907, −12.37619568684421, −12.07370757311556, −11.69822948240193, −11.05062121320427, −10.56162432625349, −9.871033483223336, −9.384402440042052, −8.642780369400909, −8.271520727366965, −7.634872053006917, −7.172185336883339, −6.545836446586927, −5.822245487514564, −5.212741704013206, −4.501375795469670, −4.165010196514281, −3.102343333887345, −2.750276648619411, −2.004753861980229, −0.7373399152547119, 0,
0.7373399152547119, 2.004753861980229, 2.750276648619411, 3.102343333887345, 4.165010196514281, 4.501375795469670, 5.212741704013206, 5.822245487514564, 6.545836446586927, 7.172185336883339, 7.634872053006917, 8.271520727366965, 8.642780369400909, 9.384402440042052, 9.871033483223336, 10.56162432625349, 11.05062121320427, 11.69822948240193, 12.07370757311556, 12.37619568684421, 13.13413735623907, 13.78972954427078, 14.35829482704737, 14.60531133268716, 15.44978779064177
