| L(s) = 1 | − 3-s − 7-s − 2·9-s + 11-s + 6·13-s + 7·17-s − 19-s + 21-s − 8·23-s + 5·27-s + 6·29-s + 4·31-s − 33-s + 8·37-s − 6·39-s − 5·41-s − 6·47-s + 49-s − 7·51-s + 4·53-s + 57-s + 4·59-s − 6·61-s + 2·63-s − 5·67-s + 8·69-s + 14·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 1.66·13-s + 1.69·17-s − 0.229·19-s + 0.218·21-s − 1.66·23-s + 0.962·27-s + 1.11·29-s + 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.960·39-s − 0.780·41-s − 0.875·47-s + 1/7·49-s − 0.980·51-s + 0.549·53-s + 0.132·57-s + 0.520·59-s − 0.768·61-s + 0.251·63-s − 0.610·67-s + 0.963·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676075041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676075041\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−16.49593969120702, −16.08319698636416, −15.47658998938845, −14.70768547734360, −14.07585307090567, −13.77139257712357, −13.01206040346460, −12.32242315418688, −11.74852132350100, −11.51101732360517, −10.55695778335856, −10.22834245085562, −9.551414670304504, −8.705516743150628, −8.207492332039304, −7.713292588565161, −6.510404560173571, −6.238993756302431, −5.743560717521542, −4.962653953951927, −4.019124709757903, −3.437461910488382, −2.677488961308999, −1.453277866508760, −0.6650906688511035,
0.6650906688511035, 1.453277866508760, 2.677488961308999, 3.437461910488382, 4.019124709757903, 4.962653953951927, 5.743560717521542, 6.238993756302431, 6.510404560173571, 7.713292588565161, 8.207492332039304, 8.705516743150628, 9.551414670304504, 10.22834245085562, 10.55695778335856, 11.51101732360517, 11.74852132350100, 12.32242315418688, 13.01206040346460, 13.77139257712357, 14.07585307090567, 14.70768547734360, 15.47658998938845, 16.08319698636416, 16.49593969120702
