| L(s) = 1 | − 2·4-s + 3·5-s − 6·11-s + 4·13-s + 4·16-s + 3·17-s − 2·19-s − 6·20-s + 6·23-s + 4·25-s + 6·29-s + 4·31-s − 7·37-s − 3·41-s − 43-s + 12·44-s + 9·47-s − 8·52-s + 6·53-s − 18·55-s + 9·59-s + 10·61-s − 8·64-s + 12·65-s − 4·67-s − 6·68-s − 2·73-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 1.80·11-s + 1.10·13-s + 16-s + 0.727·17-s − 0.458·19-s − 1.34·20-s + 1.25·23-s + 4/5·25-s + 1.11·29-s + 0.718·31-s − 1.15·37-s − 0.468·41-s − 0.152·43-s + 1.80·44-s + 1.31·47-s − 1.10·52-s + 0.824·53-s − 2.42·55-s + 1.17·59-s + 1.28·61-s − 64-s + 1.48·65-s − 0.488·67-s − 0.727·68-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630538063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630538063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.757704353671394550269388555688, −8.677781718759589525261607373664, −8.386609491702580500148378137845, −7.19528682144870768370486195193, −6.04248676207121526761624269951, −5.40863841069341263955171468202, −4.79831898195110296666259349467, −3.44521260796395376114982703678, −2.42168483312985269777609133770, −0.980257339797202376885739263498,
0.980257339797202376885739263498, 2.42168483312985269777609133770, 3.44521260796395376114982703678, 4.79831898195110296666259349467, 5.40863841069341263955171468202, 6.04248676207121526761624269951, 7.19528682144870768370486195193, 8.386609491702580500148378137845, 8.677781718759589525261607373664, 9.757704353671394550269388555688
