| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 14-s + 16-s − 4·17-s + 3·19-s − 20-s + 22-s + 6·23-s + 25-s − 28-s − 4·29-s + 7·31-s + 32-s − 4·34-s + 35-s + 8·37-s + 3·38-s − 40-s − 6·41-s − 5·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.688·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.188·28-s − 0.742·29-s + 1.25·31-s + 0.176·32-s − 0.685·34-s + 0.169·35-s + 1.31·37-s + 0.486·38-s − 0.158·40-s − 0.937·41-s − 0.762·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.51861982524619, −12.99155875272767, −12.69453977297865, −11.93173562820489, −11.66937654873507, −11.25024109576306, −10.83037862231631, −10.05388820762892, −9.849366100892791, −9.088963874391224, −8.664398978205231, −8.216221013475066, −7.455118754407085, −7.029195632511300, −6.787166499807791, −5.978256852817532, −5.720383829105132, −4.860188882475337, −4.599590909964447, −3.978182528494102, −3.447807262984128, −2.841237906944999, −2.460994148720105, −1.518705141274978, −0.9241338685677663, 0,
0.9241338685677663, 1.518705141274978, 2.460994148720105, 2.841237906944999, 3.447807262984128, 3.978182528494102, 4.599590909964447, 4.860188882475337, 5.720383829105132, 5.978256852817532, 6.787166499807791, 7.029195632511300, 7.455118754407085, 8.216221013475066, 8.664398978205231, 9.088963874391224, 9.849366100892791, 10.05388820762892, 10.83037862231631, 11.25024109576306, 11.66937654873507, 11.93173562820489, 12.69453977297865, 12.99155875272767, 13.51861982524619
