| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s − 3·9-s + 2·10-s − 11-s + 6·13-s + 16-s − 6·17-s + 3·18-s + 4·19-s − 2·20-s + 22-s − 4·23-s − 25-s − 6·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s − 3·36-s + 6·37-s − 4·38-s + 2·40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 9-s + 0.632·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 1/5·25-s − 1.17·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 1/2·36-s + 0.986·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8228574650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8228574650\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 53 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.670300669796541059758992688379, −8.737268913707583930159966316708, −8.288197940112897224885981582094, −7.58891718305931030427980967155, −6.44359756614282060066536130330, −5.84325975480226856064380217543, −4.46009110977784465199702050757, −3.48650550367727036943212305699, −2.42147208638042129045453297782, −0.74450849767196520290416116174,
0.74450849767196520290416116174, 2.42147208638042129045453297782, 3.48650550367727036943212305699, 4.46009110977784465199702050757, 5.84325975480226856064380217543, 6.44359756614282060066536130330, 7.58891718305931030427980967155, 8.288197940112897224885981582094, 8.737268913707583930159966316708, 9.670300669796541059758992688379
