| L(s) = 1 | + 3-s + 7-s + 9-s + 4·11-s + 2·13-s + 2·17-s + 21-s + 4·23-s + 27-s − 2·29-s + 8·31-s + 4·33-s + 10·37-s + 2·39-s + 2·41-s + 4·43-s − 4·47-s + 49-s + 2·51-s + 10·53-s − 4·59-s + 2·61-s + 63-s + 4·67-s + 4·69-s + 6·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.218·21-s + 0.834·23-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s + 0.702·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.528510741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.528510741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.98556934264286, −14.47782954759193, −13.97828963295777, −13.54566101674646, −12.94538256282448, −12.39310047333217, −11.77588838422654, −11.33484166654984, −10.81093002177669, −10.11298442685907, −9.450194083835704, −9.208715134151925, −8.408973061117688, −8.114377525953025, −7.393117843478716, −6.793410790596058, −6.245911472390455, −5.621337032681776, −4.812616896540726, −4.185822544820184, −3.719294808895457, −2.931350909375479, −2.301840138699133, −1.305621840928214, −0.9012603511389187,
0.9012603511389187, 1.305621840928214, 2.301840138699133, 2.931350909375479, 3.719294808895457, 4.185822544820184, 4.812616896540726, 5.621337032681776, 6.245911472390455, 6.793410790596058, 7.393117843478716, 8.114377525953025, 8.408973061117688, 9.208715134151925, 9.450194083835704, 10.11298442685907, 10.81093002177669, 11.33484166654984, 11.77588838422654, 12.39310047333217, 12.94538256282448, 13.54566101674646, 13.97828963295777, 14.47782954759193, 14.98556934264286
