| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 2·11-s + 12-s − 7·13-s + 16-s − 17-s − 18-s − 2·22-s + 4·23-s − 24-s + 7·26-s + 27-s − 4·29-s − 4·31-s − 32-s + 2·33-s + 34-s + 36-s + 4·37-s − 7·39-s − 6·41-s − 43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 1.94·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.426·22-s + 0.834·23-s − 0.204·24-s + 1.37·26-s + 0.192·27-s − 0.742·29-s − 0.718·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s − 1.12·39-s − 0.937·41-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.90174303992400, −13.13951569507145, −12.82653622748112, −12.27321755342726, −11.83878016206019, −11.29494008730175, −10.82869114489043, −10.22777147927876, −9.707943656106872, −9.348460293591275, −9.080685282980637, −8.367861068454670, −7.832249138725714, −7.485610626638464, −6.820140949584467, −6.682404981954779, −5.763960032378894, −5.159130863351742, −4.652347162095883, −4.048700864618662, −3.211637870293409, −2.915625355114021, −2.007809176559561, −1.807833312054930, −0.7818603722463423, 0,
0.7818603722463423, 1.807833312054930, 2.007809176559561, 2.915625355114021, 3.211637870293409, 4.048700864618662, 4.652347162095883, 5.159130863351742, 5.763960032378894, 6.682404981954779, 6.820140949584467, 7.485610626638464, 7.832249138725714, 8.367861068454670, 9.080685282980637, 9.348460293591275, 9.707943656106872, 10.22777147927876, 10.82869114489043, 11.29494008730175, 11.83878016206019, 12.27321755342726, 12.82653622748112, 13.13951569507145, 13.90174303992400
