| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 4·7-s − 8-s + 9-s − 3·10-s − 3·11-s + 12-s − 4·14-s + 3·15-s + 16-s − 18-s − 8·19-s + 3·20-s + 4·21-s + 3·22-s + 6·23-s − 24-s + 4·25-s + 27-s + 4·28-s + 3·29-s − 3·30-s + 7·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.904·11-s + 0.288·12-s − 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.670·20-s + 0.872·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s + 0.755·28-s + 0.557·29-s − 0.547·30-s + 1.25·31-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 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.88950124678664, −12.62200608648792, −11.97815830929758, −11.39808199236315, −10.89169710183873, −10.55323902147867, −10.18867783970471, −9.827700980059408, −9.097410993426182, −8.657463548694015, −8.511051926688242, −7.950929098583563, −7.540566194690964, −6.797314058174300, −6.536657901134762, −5.912655356445281, −5.290038379330845, −4.838211684206579, −4.569850909738155, −3.663959081474862, −2.895168591230494, −2.472006311971487, −1.975376760083953, −1.584265412205682, −1.012215905511599, 0,
1.012215905511599, 1.584265412205682, 1.975376760083953, 2.472006311971487, 2.895168591230494, 3.663959081474862, 4.569850909738155, 4.838211684206579, 5.290038379330845, 5.912655356445281, 6.536657901134762, 6.797314058174300, 7.540566194690964, 7.950929098583563, 8.511051926688242, 8.657463548694015, 9.097410993426182, 9.827700980059408, 10.18867783970471, 10.55323902147867, 10.89169710183873, 11.39808199236315, 11.97815830929758, 12.62200608648792, 12.88950124678664
