| L(s) = 1 | + 2·11-s + 2·13-s + 4·17-s + 2·23-s − 5·25-s − 6·29-s − 4·31-s − 2·37-s + 2·41-s + 4·43-s + 10·47-s − 7·49-s − 6·53-s − 4·59-s + 2·61-s − 4·67-s − 4·71-s − 6·73-s − 8·79-s − 2·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 0.554·13-s + 0.970·17-s + 0.417·23-s − 25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.45·47-s − 49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.474·71-s − 0.702·73-s − 0.900·79-s − 0.219·83-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.60072124060011, −15.04249573825473, −14.37723819769271, −14.17340646308792, −13.39960040240011, −12.93780685735055, −12.37401858156381, −11.77639778853399, −11.31687780031545, −10.73992528598867, −10.17770660398802, −9.427760629270739, −9.156653130902684, −8.460006442390492, −7.710436260104582, −7.383077797725848, −6.625723682867745, −5.844146493765576, −5.632210402501585, −4.711340650105094, −3.949706069482124, −3.530642353115742, −2.729165311544100, −1.759460917007888, −1.175272431013511, 0,
1.175272431013511, 1.759460917007888, 2.729165311544100, 3.530642353115742, 3.949706069482124, 4.711340650105094, 5.632210402501585, 5.844146493765576, 6.625723682867745, 7.383077797725848, 7.710436260104582, 8.460006442390492, 9.156653130902684, 9.427760629270739, 10.17770660398802, 10.73992528598867, 11.31687780031545, 11.77639778853399, 12.37401858156381, 12.93780685735055, 13.39960040240011, 14.17340646308792, 14.37723819769271, 15.04249573825473, 15.60072124060011
