| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 4·11-s + 12-s + 6·13-s − 2·14-s − 15-s + 16-s + 4·17-s + 18-s − 20-s − 2·21-s + 4·22-s + 4·23-s + 24-s + 25-s + 6·26-s + 27-s − 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.66·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.715086594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.715086594\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−16.26559462417002, −15.94220477729701, −15.25518298142248, −14.87910440889292, −14.10217514700244, −13.73879456184940, −13.19005119030554, −12.54471547133889, −12.01512072602714, −11.46991317682877, −10.74325332359197, −10.21745022717389, −9.311470400399279, −8.875875239952506, −8.298667398991772, −7.336051975631522, −6.993907074285125, −6.090358109586712, −5.789545261646105, −4.639232438708309, −3.924810954921145, −3.448460497880277, −2.983400728425747, −1.701957946653454, −0.9512781326397814,
0.9512781326397814, 1.701957946653454, 2.983400728425747, 3.448460497880277, 3.924810954921145, 4.639232438708309, 5.789545261646105, 6.090358109586712, 6.993907074285125, 7.336051975631522, 8.298667398991772, 8.875875239952506, 9.311470400399279, 10.21745022717389, 10.74325332359197, 11.46991317682877, 12.01512072602714, 12.54471547133889, 13.19005119030554, 13.73879456184940, 14.10217514700244, 14.87910440889292, 15.25518298142248, 15.94220477729701, 16.26559462417002
