| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s − 5·7-s + 3·8-s + 9-s − 3·11-s + 2·12-s + 5·14-s − 16-s − 5·17-s − 18-s + 4·19-s + 10·21-s + 3·22-s − 4·23-s − 6·24-s + 4·27-s + 5·28-s − 29-s − 31-s − 5·32-s + 6·33-s + 5·34-s − 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s − 1.88·7-s + 1.06·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.33·14-s − 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.917·19-s + 2.18·21-s + 0.639·22-s − 0.834·23-s − 1.22·24-s + 0.769·27-s + 0.944·28-s − 0.185·29-s − 0.179·31-s − 0.883·32-s + 1.04·33-s + 0.857·34-s − 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.002270055586695993567888496325, −7.31121847916479401724076263559, −6.47178880416182914455943306721, −5.95032165881297229049295560717, −5.17919523080184906109756286430, −4.35700078003973579981253524559, −3.40806349306029916484132306650, −2.37960135452325100204994795758, −0.71835443084071970736782441763, 0,
0.71835443084071970736782441763, 2.37960135452325100204994795758, 3.40806349306029916484132306650, 4.35700078003973579981253524559, 5.17919523080184906109756286430, 5.95032165881297229049295560717, 6.47178880416182914455943306721, 7.31121847916479401724076263559, 8.002270055586695993567888496325
