| L(s) = 1 | + 2-s + 4-s − 3·5-s − 4·7-s + 8-s − 3·10-s + 6·11-s + 2·13-s − 4·14-s + 16-s − 3·17-s + 2·19-s − 3·20-s + 6·22-s − 6·23-s + 4·25-s + 2·26-s − 4·28-s − 3·29-s + 2·31-s + 32-s − 3·34-s + 12·35-s + 2·38-s − 3·40-s − 3·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.51·7-s + 0.353·8-s − 0.948·10-s + 1.80·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.670·20-s + 1.27·22-s − 1.25·23-s + 4/5·25-s + 0.392·26-s − 0.755·28-s − 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s + 2.02·35-s + 0.324·38-s − 0.474·40-s − 0.468·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24642 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24642 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−15.53761713806184, −15.30927265487671, −14.61976314178159, −13.90930071925814, −13.61437952495963, −12.92275894692332, −12.22810710784811, −12.08382740205460, −11.43757359295792, −11.08201901386704, −10.16767887757952, −9.679492283331378, −8.994925196271814, −8.524238265966301, −7.732948291438659, −7.082243253216225, −6.585291208179776, −6.226707292619381, −5.498322770666527, −4.441884251086173, −3.908041235389482, −3.690097698569993, −3.053283777268334, −2.045783988570251, −0.9843853402178401, 0,
0.9843853402178401, 2.045783988570251, 3.053283777268334, 3.690097698569993, 3.908041235389482, 4.441884251086173, 5.498322770666527, 6.226707292619381, 6.585291208179776, 7.082243253216225, 7.732948291438659, 8.524238265966301, 8.994925196271814, 9.679492283331378, 10.16767887757952, 11.08201901386704, 11.43757359295792, 12.08382740205460, 12.22810710784811, 12.92275894692332, 13.61437952495963, 13.90930071925814, 14.61976314178159, 15.30927265487671, 15.53761713806184
