| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s + 2·7-s − 3·8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·14-s + 2·15-s − 16-s + 18-s − 2·20-s + 2·21-s − 3·24-s − 25-s − 2·26-s + 27-s − 2·28-s − 2·29-s + 2·30-s − 10·31-s + 5·32-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.516·15-s − 1/4·16-s + 0.235·18-s − 0.447·20-s + 0.436·21-s − 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.365·30-s − 1.79·31-s + 0.883·32-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.948116218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.948116218\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 71 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.69371666154320068047931531822, −11.58416538134391689105524935957, −10.24964466347711041141788937449, −9.340867901774206272924397808123, −8.538079217351861871995790864690, −7.26942572648918184499875106677, −5.77911041919239232481630021544, −4.93642128518106609960687227916, −3.66708077258385286242308094212, −2.10246999762039359182885572844,
2.10246999762039359182885572844, 3.66708077258385286242308094212, 4.93642128518106609960687227916, 5.77911041919239232481630021544, 7.26942572648918184499875106677, 8.538079217351861871995790864690, 9.340867901774206272924397808123, 10.24964466347711041141788937449, 11.58416538134391689105524935957, 12.69371666154320068047931531822
