| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 3·7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s − 3·14-s − 15-s + 16-s + 18-s − 5·19-s − 20-s − 3·21-s + 2·22-s + 24-s + 25-s + 26-s + 27-s − 3·28-s − 4·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.14·19-s − 0.223·20-s − 0.654·21-s + 0.426·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.566·28-s − 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−7.20419206267866823379293990266, −6.70556202549116875630786539124, −6.18875493247127192399322012321, −5.31770951217856997160094044000, −4.37132813709390319734889669354, −3.79094712619304313724888695832, −3.27343763127102231193761903163, −2.44897068699588892845389562958, −1.47275212150405751920376563979, 0,
1.47275212150405751920376563979, 2.44897068699588892845389562958, 3.27343763127102231193761903163, 3.79094712619304313724888695832, 4.37132813709390319734889669354, 5.31770951217856997160094044000, 6.18875493247127192399322012321, 6.70556202549116875630786539124, 7.20419206267866823379293990266
