| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s − 17-s + 4·19-s − 27-s − 2·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s + 2·41-s − 4·43-s + 8·47-s − 7·49-s + 51-s + 10·53-s − 4·57-s − 4·59-s + 14·61-s − 4·67-s + 14·73-s + 8·79-s + 81-s − 4·83-s + 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 49-s + 0.140·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.488·67-s + 1.63·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.772446743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772446743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.81910077435280, −12.25832176400049, −11.87597307604067, −11.47184366061709, −10.91420245922911, −10.54750970511065, −10.06547552282149, −9.646274265410074, −9.246354991496237, −8.446801543061947, −8.135931614478030, −7.614381908294316, −7.161453142797130, −6.648931581831634, −6.157248005391712, −5.448646123547859, −5.290998928521412, −4.728024501830431, −4.148911599950890, −3.585722937528873, −2.810079509177241, −2.505345420546098, −1.775769462760920, −0.9414557956201852, −0.4485927043714717,
0.4485927043714717, 0.9414557956201852, 1.775769462760920, 2.505345420546098, 2.810079509177241, 3.585722937528873, 4.148911599950890, 4.728024501830431, 5.290998928521412, 5.448646123547859, 6.157248005391712, 6.648931581831634, 7.161453142797130, 7.614381908294316, 8.135931614478030, 8.446801543061947, 9.246354991496237, 9.646274265410074, 10.06547552282149, 10.54750970511065, 10.91420245922911, 11.47184366061709, 11.87597307604067, 12.25832176400049, 12.81910077435280
