| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 4·11-s + 6·13-s − 14-s + 16-s + 17-s − 4·19-s − 20-s + 4·22-s + 8·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 32-s + 34-s + 35-s − 6·37-s − 4·38-s − 40-s + 6·41-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.176·32-s + 0.171·34-s + 0.169·35-s − 0.986·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.799411379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.799411379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−16.44726130839512, −15.82598320939882, −15.42721904654370, −14.80553192284660, −14.24428479946112, −13.68398104578439, −13.09609074150407, −12.54035284792863, −12.04762469288605, −11.26618598141434, −10.94109093765409, −10.36735384469233, −9.297105497397079, −8.891112809615830, −8.296556420102253, −7.399062186066743, −6.726992916480250, −6.297408716793171, −5.663961349999494, −4.707265770505789, −4.067861444950118, −3.511965855943750, −2.877498223398498, −1.652211111357431, −0.8750428892125403,
0.8750428892125403, 1.652211111357431, 2.877498223398498, 3.511965855943750, 4.067861444950118, 4.707265770505789, 5.663961349999494, 6.297408716793171, 6.726992916480250, 7.399062186066743, 8.296556420102253, 8.891112809615830, 9.297105497397079, 10.36735384469233, 10.94109093765409, 11.26618598141434, 12.04762469288605, 12.54035284792863, 13.09609074150407, 13.68398104578439, 14.24428479946112, 14.80553192284660, 15.42721904654370, 15.82598320939882, 16.44726130839512
