| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s − 3·7-s + 8-s + 9-s − 2·10-s − 6·11-s − 2·12-s + 4·13-s − 3·14-s + 4·15-s + 16-s − 4·17-s + 18-s + 3·19-s − 2·20-s + 6·21-s − 6·22-s − 4·23-s − 2·24-s − 25-s + 4·26-s + 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.577·12-s + 1.10·13-s − 0.801·14-s + 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.688·19-s − 0.447·20-s + 1.30·21-s − 1.27·22-s − 0.834·23-s − 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 131 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−11.43880062354613718244110270408, −10.94881410365669929579075243360, −9.987737144385883613813836735448, −8.378344629809609139974065194002, −7.27025426282522016089440167182, −6.18857529186169682847994848330, −5.45670365344481721710026179833, −4.19769010361100718418434941152, −2.94970678295522440004801294172, 0,
2.94970678295522440004801294172, 4.19769010361100718418434941152, 5.45670365344481721710026179833, 6.18857529186169682847994848330, 7.27025426282522016089440167182, 8.378344629809609139974065194002, 9.987737144385883613813836735448, 10.94881410365669929579075243360, 11.43880062354613718244110270408
