| L(s) = 1 | + 2-s + 3·3-s − 4-s − 5-s + 3·6-s + 2·7-s − 3·8-s + 6·9-s − 10-s − 3·12-s − 13-s + 2·14-s − 3·15-s − 16-s + 3·17-s + 6·18-s − 2·19-s + 20-s + 6·21-s + 3·23-s − 9·24-s + 25-s − 26-s + 9·27-s − 2·28-s + 5·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.755·7-s − 1.06·8-s + 2·9-s − 0.316·10-s − 0.866·12-s − 0.277·13-s + 0.534·14-s − 0.774·15-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 0.458·19-s + 0.223·20-s + 1.30·21-s + 0.625·23-s − 1.83·24-s + 1/5·25-s − 0.196·26-s + 1.73·27-s − 0.377·28-s + 0.928·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.976916387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.976916387\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.74350063602367030573920268471, −7.58835986181296175148697837555, −6.51360096390336955650130642169, −5.54983868364798826162143555264, −4.66253823121674624702234235363, −4.29364938374254802804126183259, −3.48312096216053555280166971640, −2.93060362983818886962882277117, −2.12115819758099243511560567089, −0.950551801700537359159001441602,
0.950551801700537359159001441602, 2.12115819758099243511560567089, 2.93060362983818886962882277117, 3.48312096216053555280166971640, 4.29364938374254802804126183259, 4.66253823121674624702234235363, 5.54983868364798826162143555264, 6.51360096390336955650130642169, 7.58835986181296175148697837555, 7.74350063602367030573920268471
