| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 9-s − 2·11-s + 2·13-s + 4·15-s + 17-s − 4·19-s + 4·21-s + 2·23-s − 25-s − 4·27-s + 2·29-s − 10·31-s − 4·33-s + 4·35-s + 10·37-s + 4·39-s + 2·41-s − 4·43-s + 2·45-s − 3·49-s + 2·51-s + 6·53-s − 4·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.242·17-s − 0.917·19-s + 0.872·21-s + 0.417·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.79·31-s − 0.696·33-s + 0.676·35-s + 1.64·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.436176540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.436176540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−10.71779368269295972220533446397, −9.768337257378259964663978784481, −8.961507486337233612981409401631, −8.255569150844395436462291316208, −7.48619775936454615878675223541, −6.15831786055430724443121244113, −5.23359945854130548308523257732, −3.94565925171278896665300057972, −2.68202815256222777860689778513, −1.76101461085344565548535643091,
1.76101461085344565548535643091, 2.68202815256222777860689778513, 3.94565925171278896665300057972, 5.23359945854130548308523257732, 6.15831786055430724443121244113, 7.48619775936454615878675223541, 8.255569150844395436462291316208, 8.961507486337233612981409401631, 9.768337257378259964663978784481, 10.71779368269295972220533446397
