| L(s) = 1 | − 3-s − 7-s − 2·9-s + 3·11-s − 4·13-s + 6·17-s + 2·19-s + 21-s − 9·23-s + 5·27-s + 6·29-s − 8·31-s − 3·33-s + 2·37-s + 4·39-s − 9·41-s − 2·43-s − 6·49-s − 6·51-s − 6·53-s − 2·57-s + 3·59-s + 61-s + 2·63-s − 8·67-s + 9·69-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.218·21-s − 1.87·23-s + 0.962·27-s + 1.11·29-s − 1.43·31-s − 0.522·33-s + 0.328·37-s + 0.640·39-s − 1.40·41-s − 0.304·43-s − 6/7·49-s − 0.840·51-s − 0.824·53-s − 0.264·57-s + 0.390·59-s + 0.128·61-s + 0.251·63-s − 0.977·67-s + 1.08·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.85181964224698, −13.20831957019922, −12.44165877732140, −12.15152360522942, −11.96529807017555, −11.42710781462551, −10.87651035756541, −10.18085320716760, −9.836138257228231, −9.593704798544447, −8.788240647179801, −8.363480350897183, −7.734017305160529, −7.341280541389470, −6.670945432124202, −6.145270590075050, −5.850637483924179, −5.117904358181451, −4.842301177915819, −4.000532661577274, −3.390424968608829, −3.046616703935311, −2.148304821110949, −1.567310910672007, −0.6888810750440949, 0,
0.6888810750440949, 1.567310910672007, 2.148304821110949, 3.046616703935311, 3.390424968608829, 4.000532661577274, 4.842301177915819, 5.117904358181451, 5.850637483924179, 6.145270590075050, 6.670945432124202, 7.341280541389470, 7.734017305160529, 8.363480350897183, 8.788240647179801, 9.593704798544447, 9.836138257228231, 10.18085320716760, 10.87651035756541, 11.42710781462551, 11.96529807017555, 12.15152360522942, 12.44165877732140, 13.20831957019922, 13.85181964224698
