| L(s) = 1 | − 3-s − 2·4-s + 9-s + 4·11-s + 2·12-s + 2·13-s + 4·16-s − 3·17-s + 7·19-s + 5·23-s − 27-s − 29-s − 4·33-s − 2·36-s − 8·37-s − 2·39-s + 5·41-s − 4·43-s − 8·44-s + 7·47-s − 4·48-s + 3·51-s − 4·52-s − 53-s − 7·57-s − 2·59-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 0.554·13-s + 16-s − 0.727·17-s + 1.60·19-s + 1.04·23-s − 0.192·27-s − 0.185·29-s − 0.696·33-s − 1/3·36-s − 1.31·37-s − 0.320·39-s + 0.780·41-s − 0.609·43-s − 1.20·44-s + 1.02·47-s − 0.577·48-s + 0.420·51-s − 0.554·52-s − 0.137·53-s − 0.927·57-s − 0.260·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710267953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710267953\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−13.68275974351283, −13.25490788274199, −12.75030818950192, −12.14944815329349, −11.83131513931223, −11.25084586632107, −10.83181856989420, −10.23339704943708, −9.632545081910866, −9.288855929868968, −8.750982664823389, −8.507023890426365, −7.477299291482188, −7.301398602280318, −6.548424422519609, −6.072465570336759, −5.461320943376679, −5.014480054472738, −4.486587132360428, −3.838766905422960, −3.480730947252364, −2.745029612406719, −1.609823480398435, −1.166586481976076, −0.4976776405949813,
0.4976776405949813, 1.166586481976076, 1.609823480398435, 2.745029612406719, 3.480730947252364, 3.838766905422960, 4.486587132360428, 5.014480054472738, 5.461320943376679, 6.072465570336759, 6.548424422519609, 7.301398602280318, 7.477299291482188, 8.507023890426365, 8.750982664823389, 9.288855929868968, 9.632545081910866, 10.23339704943708, 10.83181856989420, 11.25084586632107, 11.83131513931223, 12.14944815329349, 12.75030818950192, 13.25490788274199, 13.68275974351283
