| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 5·13-s + 14-s + 15-s + 16-s + 7·17-s + 18-s + 7·19-s + 20-s + 21-s − 4·22-s + 23-s + 24-s − 4·25-s − 5·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.980·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128226 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128226 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| 71 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.66924940712180, −13.38037263476927, −12.93122979103245, −12.20625962695233, −12.01682954697118, −11.54612783810541, −10.77144422010737, −10.29536469854709, −9.793167959005955, −9.630563209346041, −8.906339248905616, −8.119835732813469, −7.608812687903881, −7.495802051163650, −7.035279724427257, −5.924840510725031, −5.690997430383214, −5.086053115596420, −4.881797324342776, −4.001767276112734, −3.346040698517902, −2.943343529952100, −2.387793712002973, −1.763727031489375, −1.113831549642238, 0,
1.113831549642238, 1.763727031489375, 2.387793712002973, 2.943343529952100, 3.346040698517902, 4.001767276112734, 4.881797324342776, 5.086053115596420, 5.690997430383214, 5.924840510725031, 7.035279724427257, 7.495802051163650, 7.608812687903881, 8.119835732813469, 8.906339248905616, 9.630563209346041, 9.793167959005955, 10.29536469854709, 10.77144422010737, 11.54612783810541, 12.01682954697118, 12.20625962695233, 12.93122979103245, 13.38037263476927, 13.66924940712180
