| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s − 2·9-s + 3·10-s − 3·11-s − 12-s − 6·13-s − 14-s + 3·15-s + 16-s − 5·17-s + 2·18-s + 5·19-s − 3·20-s − 21-s + 3·22-s − 8·23-s + 24-s + 4·25-s + 6·26-s + 5·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.904·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 1.21·17-s + 0.471·18-s + 1.14·19-s − 0.670·20-s − 0.218·21-s + 0.639·22-s − 1.66·23-s + 0.204·24-s + 4/5·25-s + 1.17·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3206 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3206 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 229 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−8.121599144406585454757807722295, −7.23647193806374039641682591444, −6.76253154160635381849505370308, −5.41871038981560489529134245702, −5.05307294473980732430810443088, −3.96197042669907703296377652947, −2.93792311141220772085794610168, −1.97246834137703663349024105293, 0, 0,
1.97246834137703663349024105293, 2.93792311141220772085794610168, 3.96197042669907703296377652947, 5.05307294473980732430810443088, 5.41871038981560489529134245702, 6.76253154160635381849505370308, 7.23647193806374039641682591444, 8.121599144406585454757807722295
