| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 5·13-s − 3·14-s − 15-s + 16-s + 18-s − 4·19-s + 20-s + 3·21-s − 22-s − 23-s − 24-s + 25-s + 5·26-s − 27-s − 3·28-s + 4·29-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 − 0.301·11-s − 0.288·12-s + 1.38·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s − 0.566·28-s + 0.742·29-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 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−7.04392260235960685083763314849, −6.56000182535802023829593748232, −5.99863132455249035181980146364, −5.53740617385996787080653874853, −4.65948754077714757270496129024, −3.81122454192251393306094806152, −3.28366495019125636704007048405, −2.28843781489736289359790223804, −1.34051106490718409516918299306, 0,
1.34051106490718409516918299306, 2.28843781489736289359790223804, 3.28366495019125636704007048405, 3.81122454192251393306094806152, 4.65948754077714757270496129024, 5.53740617385996787080653874853, 5.99863132455249035181980146364, 6.56000182535802023829593748232, 7.04392260235960685083763314849
