| 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
| \(L(1)\) |
\(\approx\) |
\(3.434751160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.434751160\) |
| \(L(\frac{3}{2})\) |
|
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.ac |
| 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.d |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 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.j |
| 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
−13.49883877616358, −13.07550293472000, −12.92342817072002, −12.33822300359361, −11.92012858788794, −11.35597633368490, −10.77105945314884, −10.66641602832564, −9.826608954376479, −9.364343386558607, −8.676448734516088, −8.158498329519941, −7.397357275731741, −7.093623426817022, −6.414863259863609, −5.888973532055917, −5.459434668223763, −5.020380170563234, −4.530975261013021, −4.018615486395350, −3.285210117942994, −2.737547378569577, −2.330480836046648, −1.268234395789164, −0.4839215376906436,
0.4839215376906436, 1.268234395789164, 2.330480836046648, 2.737547378569577, 3.285210117942994, 4.018615486395350, 4.530975261013021, 5.020380170563234, 5.459434668223763, 5.888973532055917, 6.414863259863609, 7.093623426817022, 7.397357275731741, 8.158498329519941, 8.676448734516088, 9.364343386558607, 9.826608954376479, 10.66641602832564, 10.77105945314884, 11.35597633368490, 11.92012858788794, 12.33822300359361, 12.92342817072002, 13.07550293472000, 13.49883877616358
