| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s + 11-s − 12-s + 2·13-s + 14-s + 15-s + 16-s + 3·17-s + 2·18-s + 19-s − 20-s + 21-s − 22-s + 6·23-s + 24-s + 25-s − 2·26-s + 5·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8583287974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8583287974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.87214666248272, −13.19391349853648, −12.78088383838921, −12.20471255573920, −11.73771869206652, −11.31067978836395, −10.97393668808721, −10.41600958461502, −9.882393009317732, −9.338990753046999, −8.747837781783610, −8.482971808829110, −7.822625232263007, −7.235268653165628, −6.752973262209269, −6.315159842571991, −5.587641543502655, −5.294055681562890, −4.560600718015256, −3.659520898345399, −3.290907209415508, −2.722784328365730, −1.761060233263416, −1.061450745975199, −0.4013087446222819,
0.4013087446222819, 1.061450745975199, 1.761060233263416, 2.722784328365730, 3.290907209415508, 3.659520898345399, 4.560600718015256, 5.294055681562890, 5.587641543502655, 6.315159842571991, 6.752973262209269, 7.235268653165628, 7.822625232263007, 8.482971808829110, 8.747837781783610, 9.338990753046999, 9.882393009317732, 10.41600958461502, 10.97393668808721, 11.31067978836395, 11.73771869206652, 12.20471255573920, 12.78088383838921, 13.19391349853648, 13.87214666248272
