| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 2·7-s − 8-s + 9-s − 2·10-s + 12-s + 2·14-s + 2·15-s + 16-s − 2·17-s − 18-s + 6·19-s + 2·20-s − 2·21-s − 4·23-s − 24-s − 25-s + 27-s − 2·28-s + 10·29-s − 2·30-s + 10·31-s − 32-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 − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.37·19-s + 0.447·20-s − 0.436·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.85·29-s − 0.365·30-s + 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.445807482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.445807482\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.64192722758062, −13.30130750404792, −12.37655741247685, −12.14491069403626, −11.75317597661709, −10.92273353584705, −10.33518046916626, −9.988971361333766, −9.783539077087777, −9.143713314988308, −8.747715540829970, −8.207475287610878, −7.742140850770861, −7.048738524014543, −6.644912784482792, −6.154485688724699, −5.690553469650790, −4.917867732279939, −4.400750977635713, −3.502773282818392, −3.049511128616273, −2.550587998505590, −1.901240648219038, −1.272269351775662, −0.5250148889220195,
0.5250148889220195, 1.272269351775662, 1.901240648219038, 2.550587998505590, 3.049511128616273, 3.502773282818392, 4.400750977635713, 4.917867732279939, 5.690553469650790, 6.154485688724699, 6.644912784482792, 7.048738524014543, 7.742140850770861, 8.207475287610878, 8.747715540829970, 9.143713314988308, 9.783539077087777, 9.988971361333766, 10.33518046916626, 10.92273353584705, 11.75317597661709, 12.14491069403626, 12.37655741247685, 13.30130750404792, 13.64192722758062
