| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s − 4·11-s − 12-s − 4·14-s + 15-s + 16-s − 2·17-s + 18-s + 19-s − 20-s + 4·21-s − 4·22-s − 4·23-s − 24-s + 25-s − 27-s − 4·28-s + 2·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 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.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.84830446436044, −13.31735337879536, −13.17838809415503, −12.56650428144053, −12.16556762110053, −11.71691517480544, −11.22787964987157, −10.50366971464385, −10.24120301390407, −9.805493872348536, −9.183110753185695, −8.394101783298146, −7.966104109422866, −7.301434669214126, −6.877225047021718, −6.251743863326084, −5.975158390192316, −5.379623498226549, −4.588953672516228, −4.384219545174290, −3.541402359249953, −2.972532208562439, −2.638879347494970, −1.717152569043016, −0.6569558859031617, 0,
0.6569558859031617, 1.717152569043016, 2.638879347494970, 2.972532208562439, 3.541402359249953, 4.384219545174290, 4.588953672516228, 5.379623498226549, 5.975158390192316, 6.251743863326084, 6.877225047021718, 7.301434669214126, 7.966104109422866, 8.394101783298146, 9.183110753185695, 9.805493872348536, 10.24120301390407, 10.50366971464385, 11.22787964987157, 11.71691517480544, 12.16556762110053, 12.56650428144053, 13.17838809415503, 13.31735337879536, 13.84830446436044
