| L(s) = 1 | + 2-s + 4-s + 8-s − 6·13-s + 16-s − 6·17-s + 6·19-s + 6·23-s − 6·26-s − 6·29-s + 4·31-s + 32-s − 6·34-s + 2·37-s + 6·38-s − 6·41-s − 6·43-s + 6·46-s + 6·47-s − 7·49-s − 6·52-s − 12·53-s − 6·58-s + 12·59-s + 6·61-s + 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.66·13-s + 1/4·16-s − 1.45·17-s + 1.37·19-s + 1.25·23-s − 1.17·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.973·38-s − 0.937·41-s − 0.914·43-s + 0.884·46-s + 0.875·47-s − 49-s − 0.832·52-s − 1.64·53-s − 0.787·58-s + 1.56·59-s + 0.768·61-s + 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.74152787657212, −14.14812452409993, −13.61980584102549, −13.17583348970797, −12.71286896088234, −12.20788390118889, −11.51316047636743, −11.37610699955251, −10.70685972741240, −9.972996633911972, −9.599022071730238, −9.085787761925486, −8.378125057242573, −7.653144901634002, −7.289014570526736, −6.691572841070852, −6.305002204535672, −5.253692977284114, −5.060997706156209, −4.625065683097846, −3.723527594937140, −3.208491907291758, −2.459451577456549, −2.029720025250196, −1.007059892685340, 0,
1.007059892685340, 2.029720025250196, 2.459451577456549, 3.208491907291758, 3.723527594937140, 4.625065683097846, 5.060997706156209, 5.253692977284114, 6.305002204535672, 6.691572841070852, 7.289014570526736, 7.653144901634002, 8.378125057242573, 9.085787761925486, 9.599022071730238, 9.972996633911972, 10.70685972741240, 11.37610699955251, 11.51316047636743, 12.20788390118889, 12.71286896088234, 13.17583348970797, 13.61980584102549, 14.14812452409993, 14.74152787657212
