| L(s) = 1 | − 2·3-s + 7-s + 9-s − 4·11-s − 4·13-s + 17-s − 6·19-s − 2·21-s + 4·27-s − 6·29-s − 4·31-s + 8·33-s − 10·37-s + 8·39-s + 6·41-s + 4·47-s + 49-s − 2·51-s + 14·53-s + 12·57-s − 6·59-s + 12·61-s + 63-s − 4·67-s + 8·71-s − 2·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.242·17-s − 1.37·19-s − 0.436·21-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 1.39·33-s − 1.64·37-s + 1.28·39-s + 0.937·41-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 1.92·53-s + 1.58·57-s − 0.781·59-s + 1.53·61-s + 0.125·63-s − 0.488·67-s + 0.949·71-s − 0.234·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.37592946256359, −12.98534037364051, −12.59018245980382, −12.05701906691474, −11.85448681393270, −11.06852066975061, −10.78272840638904, −10.50766294251171, −9.976794386865682, −9.410686995604592, −8.743786896128740, −8.380746879235031, −7.721105480832710, −7.271464288657213, −6.892011471454383, −6.235091220149818, −5.622209613399387, −5.263490218530630, −5.073200519345249, −4.225495434138696, −3.887778364273584, −2.920998916651686, −2.368889003711458, −1.925099202768793, −0.9912016526888413, 0, 0,
0.9912016526888413, 1.925099202768793, 2.368889003711458, 2.920998916651686, 3.887778364273584, 4.225495434138696, 5.073200519345249, 5.263490218530630, 5.622209613399387, 6.235091220149818, 6.892011471454383, 7.271464288657213, 7.721105480832710, 8.380746879235031, 8.743786896128740, 9.410686995604592, 9.976794386865682, 10.50766294251171, 10.78272840638904, 11.06852066975061, 11.85448681393270, 12.05701906691474, 12.59018245980382, 12.98534037364051, 13.37592946256359
