| L(s) = 1 | + 3-s − 7-s + 9-s + 3·11-s + 2·13-s + 4·17-s + 2·19-s − 21-s + 9·23-s + 27-s + 7·29-s + 10·31-s + 3·33-s + 5·37-s + 2·39-s − 8·41-s − 7·43-s + 10·47-s + 49-s + 4·51-s + 10·53-s + 2·57-s + 6·59-s + 8·61-s − 63-s − 13·67-s + 9·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.554·13-s + 0.970·17-s + 0.458·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s + 1.29·29-s + 1.79·31-s + 0.522·33-s + 0.821·37-s + 0.320·39-s − 1.24·41-s − 1.06·43-s + 1.45·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s + 0.264·57-s + 0.781·59-s + 1.02·61-s − 0.125·63-s − 1.58·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.390880379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.390880379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.96914552480740, −14.51356336844942, −13.87887263934233, −13.47386101453575, −13.07993134853987, −12.30217563261240, −11.78066836366398, −11.54198963655381, −10.47848755517819, −10.23839323878630, −9.622915742587802, −8.986922674375828, −8.595130210297279, −8.084416234347939, −7.257599668582351, −6.839052875370789, −6.300106231143906, −5.582752319484975, −4.857571297062471, −4.255075400696543, −3.470442415047950, −3.062529344763768, −2.394371390101268, −1.142347341601363, −0.9777676854875983,
0.9777676854875983, 1.142347341601363, 2.394371390101268, 3.062529344763768, 3.470442415047950, 4.255075400696543, 4.857571297062471, 5.582752319484975, 6.300106231143906, 6.839052875370789, 7.257599668582351, 8.084416234347939, 8.595130210297279, 8.986922674375828, 9.622915742587802, 10.23839323878630, 10.47848755517819, 11.54198963655381, 11.78066836366398, 12.30217563261240, 13.07993134853987, 13.47386101453575, 13.87887263934233, 14.51356336844942, 14.96914552480740
