| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s + 3·11-s + 4·13-s − 3·14-s + 16-s − 17-s − 5·19-s − 3·22-s − 4·23-s − 4·26-s + 3·28-s + 7·31-s − 32-s + 34-s + 3·37-s + 5·38-s − 2·41-s − 43-s + 3·44-s + 4·46-s − 3·47-s + 2·49-s + 4·52-s + 11·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s + 0.904·11-s + 1.10·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 1.14·19-s − 0.639·22-s − 0.834·23-s − 0.784·26-s + 0.566·28-s + 1.25·31-s − 0.176·32-s + 0.171·34-s + 0.493·37-s + 0.811·38-s − 0.312·41-s − 0.152·43-s + 0.452·44-s + 0.589·46-s − 0.437·47-s + 2/7·49-s + 0.554·52-s + 1.51·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.901005974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.901005974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.116409958143590889404439702059, −7.29790639067538548649066958979, −6.39085690022613470657946913654, −6.13389636621518824658213717518, −5.03008463493375866752233662183, −4.24711973388829483710341644421, −3.61582155028699933045796673829, −2.36308918593529719970499566945, −1.66974386410648720345648366017, −0.811949523662818008792878601116,
0.811949523662818008792878601116, 1.66974386410648720345648366017, 2.36308918593529719970499566945, 3.61582155028699933045796673829, 4.24711973388829483710341644421, 5.03008463493375866752233662183, 6.13389636621518824658213717518, 6.39085690022613470657946913654, 7.29790639067538548649066958979, 8.116409958143590889404439702059
