| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·11-s + 13-s + 14-s + 16-s + 19-s + 3·22-s + 3·23-s − 26-s − 28-s + 6·29-s − 7·31-s − 32-s + 10·37-s − 38-s − 12·41-s + 43-s − 3·44-s − 3·46-s + 6·47-s + 49-s + 52-s + 9·53-s + 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s + 0.639·22-s + 0.625·23-s − 0.196·26-s − 0.188·28-s + 1.11·29-s − 1.25·31-s − 0.176·32-s + 1.64·37-s − 0.162·38-s − 1.87·41-s + 0.152·43-s − 0.452·44-s − 0.442·46-s + 0.875·47-s + 1/7·49-s + 0.138·52-s + 1.23·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403158925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403158925\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.49665270078746, −13.62382803801548, −13.32777060437136, −12.80372634060741, −12.27197240195221, −11.66986901935918, −11.18940548740715, −10.68886479084250, −10.05829073282246, −9.906963648326944, −9.058930505773809, −8.688361635081008, −8.177111589821823, −7.531963323282728, −7.105648694714745, −6.543374873654415, −5.902526188209754, −5.337402514849813, −4.801692542409560, −3.902326570624352, −3.345478693958162, −2.610758117059717, −2.159363891526662, −1.139594040731655, −0.5052681892213681,
0.5052681892213681, 1.139594040731655, 2.159363891526662, 2.610758117059717, 3.345478693958162, 3.902326570624352, 4.801692542409560, 5.337402514849813, 5.902526188209754, 6.543374873654415, 7.105648694714745, 7.531963323282728, 8.177111589821823, 8.688361635081008, 9.058930505773809, 9.906963648326944, 10.05829073282246, 10.68886479084250, 11.18940548740715, 11.66986901935918, 12.27197240195221, 12.80372634060741, 13.32777060437136, 13.62382803801548, 14.49665270078746
