| L(s) = 1 | − 2·5-s − 3·9-s + 2·13-s + 4·17-s − 6·19-s + 4·23-s − 25-s − 10·29-s − 2·31-s − 10·37-s − 12·41-s + 4·43-s + 6·45-s + 10·53-s + 12·59-s − 4·65-s + 8·67-s − 8·71-s − 6·73-s − 79-s + 9·81-s − 10·83-s − 8·85-s + 2·89-s + 12·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 0.834·23-s − 1/5·25-s − 1.85·29-s − 0.359·31-s − 1.64·37-s − 1.87·41-s + 0.609·43-s + 0.894·45-s + 1.37·53-s + 1.56·59-s − 0.496·65-s + 0.977·67-s − 0.949·71-s − 0.702·73-s − 0.112·79-s + 81-s − 1.09·83-s − 0.867·85-s + 0.211·89-s + 1.23·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.92687218091335, −12.76831200704788, −11.99413421699404, −11.65204533484330, −11.38863128435514, −10.85606850308399, −10.31496775896465, −10.05015556659473, −9.077165196257049, −8.883338068848422, −8.434528198488318, −8.036850497656339, −7.358677887361250, −7.114540354424753, −6.432986186358958, −5.901538893797719, −5.298145288835002, −5.152075110251597, −4.148410495206765, −3.759062259536499, −3.453386510782366, −2.791803919954618, −2.064722617362020, −1.529202505467616, −0.5743701571920065, 0,
0.5743701571920065, 1.529202505467616, 2.064722617362020, 2.791803919954618, 3.453386510782366, 3.759062259536499, 4.148410495206765, 5.152075110251597, 5.298145288835002, 5.901538893797719, 6.432986186358958, 7.114540354424753, 7.358677887361250, 8.036850497656339, 8.434528198488318, 8.883338068848422, 9.077165196257049, 10.05015556659473, 10.31496775896465, 10.85606850308399, 11.38863128435514, 11.65204533484330, 11.99413421699404, 12.76831200704788, 12.92687218091335
