| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 8-s + 9-s + 4·10-s + 2·11-s − 12-s − 4·15-s + 16-s + 18-s − 2·19-s + 4·20-s + 2·22-s − 24-s + 11·25-s − 27-s + 2·29-s − 4·30-s + 8·31-s + 32-s − 2·33-s + 36-s − 10·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s − 1.03·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.894·20-s + 0.426·22-s − 0.204·24-s + 11/5·25-s − 0.192·27-s + 0.371·29-s − 0.730·30-s + 1.43·31-s + 0.176·32-s − 0.348·33-s + 1/6·36-s − 1.64·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.94829089891356, −12.43582204993835, −12.17481408068063, −11.70479032472881, −11.05154708595377, −10.62860820331711, −10.19557838952519, −9.962000282937496, −9.300609719333538, −8.893730185818114, −8.402326593445977, −7.720610402183581, −6.953677110488269, −6.642821676155321, −6.333581289038991, −5.851658479475172, −5.359605519079518, −4.952088728743191, −4.512112729370128, −3.832492385329740, −3.211672514759301, −2.572802390345100, −2.118772789411366, −1.422433895539580, −1.151273875333491, 0,
1.151273875333491, 1.422433895539580, 2.118772789411366, 2.572802390345100, 3.211672514759301, 3.832492385329740, 4.512112729370128, 4.952088728743191, 5.359605519079518, 5.851658479475172, 6.333581289038991, 6.642821676155321, 6.953677110488269, 7.720610402183581, 8.402326593445977, 8.893730185818114, 9.300609719333538, 9.962000282937496, 10.19557838952519, 10.62860820331711, 11.05154708595377, 11.70479032472881, 12.17481408068063, 12.43582204993835, 12.94829089891356
