| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 13-s + 14-s + 16-s − 2·17-s − 7·19-s + 20-s + 4·23-s + 25-s + 26-s + 28-s − 2·29-s − 3·31-s + 32-s − 2·34-s + 35-s + 2·37-s − 7·38-s + 40-s + 10·41-s − 13·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.60·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.371·29-s − 0.538·31-s + 0.176·32-s − 0.342·34-s + 0.169·35-s + 0.328·37-s − 1.13·38-s + 0.158·40-s + 1.56·41-s − 1.98·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 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
−13.52326814195836, −13.12739197057085, −12.80545731562286, −12.41425352751195, −11.68675216634548, −11.14358442783667, −10.94427482570011, −10.49934085006746, −9.694476081152284, −9.439218455919961, −8.648595115416052, −8.336918528776063, −7.796779029214721, −7.051531149971136, −6.596386121829153, −6.312205455232042, −5.581937512769678, −5.099026586200018, −4.704129883295371, −3.906872837065825, −3.713494671379577, −2.718326407355846, −2.321316156836392, −1.729771597950872, −1.024734760858728, 0,
1.024734760858728, 1.729771597950872, 2.321316156836392, 2.718326407355846, 3.713494671379577, 3.906872837065825, 4.704129883295371, 5.099026586200018, 5.581937512769678, 6.312205455232042, 6.596386121829153, 7.051531149971136, 7.796779029214721, 8.336918528776063, 8.648595115416052, 9.439218455919961, 9.694476081152284, 10.49934085006746, 10.94427482570011, 11.14358442783667, 11.68675216634548, 12.41425352751195, 12.80545731562286, 13.12739197057085, 13.52326814195836
