| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 3·11-s − 4·13-s + 15-s − 7·17-s + 3·21-s + 23-s − 4·25-s − 27-s − 4·29-s + 10·31-s − 3·33-s + 3·35-s − 2·37-s + 4·39-s − 10·41-s + 43-s − 45-s + 7·47-s + 2·49-s + 7·51-s − 2·53-s − 3·55-s + 2·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s + 0.258·15-s − 1.69·17-s + 0.654·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s − 0.742·29-s + 1.79·31-s − 0.522·33-s + 0.507·35-s − 0.328·37-s + 0.640·39-s − 1.56·41-s + 0.152·43-s − 0.149·45-s + 1.02·47-s + 2/7·49-s + 0.980·51-s − 0.274·53-s − 0.404·55-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 398544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 398544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3266324111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3266324111\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−12.33926383529466, −12.04279347763566, −11.56620202633672, −11.19922894298288, −10.66699651820091, −10.11881611924943, −9.769330943829394, −9.326615087899518, −8.944102555242554, −8.389692997152640, −7.785181377153441, −7.310783995345384, −6.704885973906500, −6.557297002234315, −6.193537638688529, −5.414273422922072, −4.964259454278442, −4.410484225252943, −3.997054909144861, −3.525324619389314, −2.861603274951265, −2.300274144041143, −1.755245152458984, −0.8734176452049560, −0.1838861475615287,
0.1838861475615287, 0.8734176452049560, 1.755245152458984, 2.300274144041143, 2.861603274951265, 3.525324619389314, 3.997054909144861, 4.410484225252943, 4.964259454278442, 5.414273422922072, 6.193537638688529, 6.557297002234315, 6.704885973906500, 7.310783995345384, 7.785181377153441, 8.389692997152640, 8.944102555242554, 9.326615087899518, 9.769330943829394, 10.11881611924943, 10.66699651820091, 11.19922894298288, 11.56620202633672, 12.04279347763566, 12.33926383529466
