| L(s) = 1 | − 11-s + 4·13-s + 2·17-s + 6·19-s − 2·23-s − 5·25-s − 10·29-s + 8·31-s + 2·37-s + 6·41-s + 8·43-s + 6·47-s − 7·49-s + 4·53-s − 6·59-s − 12·61-s − 67-s + 14·71-s − 10·73-s − 16·79-s + 4·83-s + 14·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s − 0.417·23-s − 25-s − 1.85·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s − 49-s + 0.549·53-s − 0.781·59-s − 1.53·61-s − 0.122·67-s + 1.66·71-s − 1.17·73-s − 1.80·79-s + 0.439·83-s + 1.48·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 67 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.81980175095254, −13.49648834857967, −13.12631166245013, −12.33109758305356, −12.07223471205926, −11.46627490750112, −10.99118022993783, −10.65208135753146, −9.885580089930570, −9.496199687126044, −9.167649347126652, −8.385889935584468, −7.862878736387690, −7.601997425466206, −7.000428685719397, −6.199355159004615, −5.738862174244959, −5.555263445674192, −4.625920406876524, −4.094172007703154, −3.562957716954866, −2.970604476072904, −2.332002330386410, −1.476294779816296, −0.9919211064175370, 0,
0.9919211064175370, 1.476294779816296, 2.332002330386410, 2.970604476072904, 3.562957716954866, 4.094172007703154, 4.625920406876524, 5.555263445674192, 5.738862174244959, 6.199355159004615, 7.000428685719397, 7.601997425466206, 7.862878736387690, 8.385889935584468, 9.167649347126652, 9.496199687126044, 9.885580089930570, 10.65208135753146, 10.99118022993783, 11.46627490750112, 12.07223471205926, 12.33109758305356, 13.12631166245013, 13.49648834857967, 13.81980175095254
