| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 3·11-s − 13-s + 16-s + 3·17-s + 19-s + 20-s + 3·22-s + 2·23-s − 4·25-s − 26-s + 2·29-s − 31-s + 32-s + 3·34-s − 2·37-s + 38-s + 40-s − 4·43-s + 3·44-s + 2·46-s + 7·47-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s + 0.223·20-s + 0.639·22-s + 0.417·23-s − 4/5·25-s − 0.196·26-s + 0.371·29-s − 0.179·31-s + 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.162·38-s + 0.158·40-s − 0.609·43-s + 0.452·44-s + 0.294·46-s + 1.02·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.474365643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474365643\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−10.86854857465430386588017560187, −9.918870461951558273250274050648, −9.159763668299275927660127090801, −7.963435050983651815539533078446, −6.97642119982859474551829060741, −6.09225726329028835226509898006, −5.21225146363139569251885813175, −4.10239034138391675224069292349, −3.00366413757856607909855265823, −1.57085349073372643096848893609,
1.57085349073372643096848893609, 3.00366413757856607909855265823, 4.10239034138391675224069292349, 5.21225146363139569251885813175, 6.09225726329028835226509898006, 6.97642119982859474551829060741, 7.963435050983651815539533078446, 9.159763668299275927660127090801, 9.918870461951558273250274050648, 10.86854857465430386588017560187
