| L(s) = 1 | − 5-s − 11-s − 6·13-s + 8·17-s + 4·19-s − 7·23-s + 25-s + 9·29-s + 6·31-s − 2·37-s − 3·41-s + 43-s − 8·47-s + 2·53-s + 55-s − 6·59-s − 5·61-s + 6·65-s − 67-s + 4·71-s + 8·73-s − 10·79-s + 9·83-s − 8·85-s − 89-s − 4·95-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s + 1.94·17-s + 0.917·19-s − 1.45·23-s + 1/5·25-s + 1.67·29-s + 1.07·31-s − 0.328·37-s − 0.468·41-s + 0.152·43-s − 1.16·47-s + 0.274·53-s + 0.134·55-s − 0.781·59-s − 0.640·61-s + 0.744·65-s − 0.122·67-s + 0.474·71-s + 0.936·73-s − 1.12·79-s + 0.987·83-s − 0.867·85-s − 0.105·89-s − 0.410·95-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.38749019694449, −12.19829644093530, −11.90652166810230, −11.60922887652295, −10.78042807595705, −10.24332124525803, −10.01709775962468, −9.702272025775343, −9.189021728114398, −8.367964730432307, −8.020634460052280, −7.776163451669861, −7.314751162550557, −6.720359965214516, −6.287993358284443, −5.572782835257995, −5.252478144642129, −4.715154872987442, −4.346302644319334, −3.566436975024645, −3.096180067981708, −2.746193149154368, −2.050711010224409, −1.324379221584605, −0.7184441581414923, 0,
0.7184441581414923, 1.324379221584605, 2.050711010224409, 2.746193149154368, 3.096180067981708, 3.566436975024645, 4.346302644319334, 4.715154872987442, 5.252478144642129, 5.572782835257995, 6.287993358284443, 6.720359965214516, 7.314751162550557, 7.776163451669861, 8.020634460052280, 8.367964730432307, 9.189021728114398, 9.702272025775343, 10.01709775962468, 10.24332124525803, 10.78042807595705, 11.60922887652295, 11.90652166810230, 12.19829644093530, 12.38749019694449
