| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 11-s + 12-s + 13-s − 2·14-s + 16-s + 4·17-s − 18-s + 19-s + 2·21-s + 22-s + 23-s − 24-s − 26-s + 27-s + 2·28-s − 9·29-s + 3·31-s − 32-s − 33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s + 0.436·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.67·29-s + 0.538·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743895178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743895178\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−9.394060326197841698248702732723, −8.537638379467561859664400355794, −7.75947724847373495220663686984, −7.46978861737166408888698818411, −6.21449230466502714506573926069, −5.36748707321625929087646916500, −4.26058028072949363648343209841, −3.19461026218671026134723225707, −2.16544976977216524243966688268, −1.05824878957084615629267235820,
1.05824878957084615629267235820, 2.16544976977216524243966688268, 3.19461026218671026134723225707, 4.26058028072949363648343209841, 5.36748707321625929087646916500, 6.21449230466502714506573926069, 7.46978861737166408888698818411, 7.75947724847373495220663686984, 8.537638379467561859664400355794, 9.394060326197841698248702732723
