| L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·13-s + 8·17-s − 2·21-s − 23-s − 5·25-s − 27-s − 2·29-s − 4·31-s − 6·37-s + 2·39-s − 10·41-s − 6·43-s − 3·49-s − 8·51-s − 12·53-s + 4·59-s − 10·61-s + 2·63-s − 6·67-s + 69-s + 2·73-s + 5·75-s − 6·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.94·17-s − 0.436·21-s − 0.208·23-s − 25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.986·37-s + 0.320·39-s − 1.56·41-s − 0.914·43-s − 3/7·49-s − 1.12·51-s − 1.64·53-s + 0.520·59-s − 1.28·61-s + 0.251·63-s − 0.733·67-s + 0.120·69-s + 0.234·73-s + 0.577·75-s − 0.675·79-s + 1/9·81-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
| \(L(1)\) |
\(\approx\) |
\(0.5449583410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5449583410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.26354961497634, −12.07899431940263, −11.46612809680372, −11.28995848307764, −10.55542166960030, −10.23985366717784, −9.777057730140619, −9.481885306493502, −8.734025312917460, −8.281533953068271, −7.782222952682900, −7.463966799470973, −7.058905571966491, −6.276756472366006, −5.997570781444931, −5.315188873880454, −5.071140524018755, −4.684402013098421, −3.865699395014191, −3.468067257842802, −3.002784947155731, −2.070477565569722, −1.608849016525959, −1.248149542433031, −0.1936649652384394,
0.1936649652384394, 1.248149542433031, 1.608849016525959, 2.070477565569722, 3.002784947155731, 3.468067257842802, 3.865699395014191, 4.684402013098421, 5.071140524018755, 5.315188873880454, 5.997570781444931, 6.276756472366006, 7.058905571966491, 7.463966799470973, 7.782222952682900, 8.281533953068271, 8.734025312917460, 9.481885306493502, 9.777057730140619, 10.23985366717784, 10.55542166960030, 11.28995848307764, 11.46612809680372, 12.07899431940263, 12.26354961497634
