| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 11-s − 12-s − 2·13-s − 14-s + 16-s + 6·17-s + 18-s − 4·19-s + 21-s − 22-s − 24-s − 2·26-s − 27-s − 28-s − 6·29-s − 4·31-s + 32-s + 33-s + 6·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.213·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.174·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163956351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163956351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.40217581695404, −15.92235247861064, −15.20387105418777, −14.69501082427704, −14.28363051352073, −13.44251789419850, −12.88843362645135, −12.52857585458680, −11.94927453206381, −11.31852430363938, −10.78090099574417, −9.964169105721578, −9.815953442362729, −8.728599195085854, −8.055038241971842, −7.218879546776753, −6.920282042103928, −6.003382967334737, −5.451373351938033, −5.060805429278582, −4.000246397237465, −3.595208414334346, −2.589176751144067, −1.819719844431265, −0.6101145848493949,
0.6101145848493949, 1.819719844431265, 2.589176751144067, 3.595208414334346, 4.000246397237465, 5.060805429278582, 5.451373351938033, 6.003382967334737, 6.920282042103928, 7.218879546776753, 8.055038241971842, 8.728599195085854, 9.815953442362729, 9.964169105721578, 10.78090099574417, 11.31852430363938, 11.94927453206381, 12.52857585458680, 12.88843362645135, 13.44251789419850, 14.28363051352073, 14.69501082427704, 15.20387105418777, 15.92235247861064, 16.40217581695404
