| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s − 4·11-s − 2·12-s − 4·16-s + 3·17-s − 2·18-s + 3·19-s + 8·22-s − 9·23-s − 27-s + 29-s + 8·32-s + 4·33-s − 6·34-s + 2·36-s − 8·37-s − 6·38-s − 5·41-s + 6·43-s − 8·44-s + 18·46-s + 9·47-s + 4·48-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 16-s + 0.727·17-s − 0.471·18-s + 0.688·19-s + 1.70·22-s − 1.87·23-s − 0.192·27-s + 0.185·29-s + 1.41·32-s + 0.696·33-s − 1.02·34-s + 1/3·36-s − 1.31·37-s − 0.973·38-s − 0.780·41-s + 0.914·43-s − 1.20·44-s + 2.65·46-s + 1.31·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.87072140787619, −13.53526872767674, −12.81661846621999, −12.27597037985770, −11.89168875052348, −11.41341366590877, −10.69169175845644, −10.40225130279681, −10.12432290366572, −9.552523271673888, −9.090640146052437, −8.370375768845066, −8.023260464785403, −7.567177459366905, −7.158392949946107, −6.520295509314643, −5.809050762294371, −5.447850176138007, −4.825622930105527, −4.163084807167446, −3.473839557546313, −2.671287312310410, −2.037927607792697, −1.425227730426202, −0.6312500475446820, 0,
0.6312500475446820, 1.425227730426202, 2.037927607792697, 2.671287312310410, 3.473839557546313, 4.163084807167446, 4.825622930105527, 5.447850176138007, 5.809050762294371, 6.520295509314643, 7.158392949946107, 7.567177459366905, 8.023260464785403, 8.370375768845066, 9.090640146052437, 9.552523271673888, 10.12432290366572, 10.40225130279681, 10.69169175845644, 11.41341366590877, 11.89168875052348, 12.27597037985770, 12.81661846621999, 13.53526872767674, 13.87072140787619
