| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 15-s + 16-s − 2·17-s + 18-s − 4·19-s + 20-s + 22-s + 24-s + 25-s + 26-s + 27-s + 2·29-s + 30-s − 8·31-s + 32-s + 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.23713177169034, −12.91371819942394, −12.45492582298562, −11.98805358814459, −11.37109004036325, −10.94233913119804, −10.45819105770835, −10.11276388681581, −9.310076144254955, −9.088541494228327, −8.551761364064792, −8.028949349908138, −7.488931595813726, −6.852479102989132, −6.578663602819665, −6.037013389556161, −5.407158937949124, −5.017330520673826, −4.272332629136279, −3.948822454217758, −3.372531575436240, −2.779643120695681, −2.128910734370703, −1.791582844742574, −1.034215437628039, 0,
1.034215437628039, 1.791582844742574, 2.128910734370703, 2.779643120695681, 3.372531575436240, 3.948822454217758, 4.272332629136279, 5.017330520673826, 5.407158937949124, 6.037013389556161, 6.578663602819665, 6.852479102989132, 7.488931595813726, 8.028949349908138, 8.551761364064792, 9.088541494228327, 9.310076144254955, 10.11276388681581, 10.45819105770835, 10.94233913119804, 11.37109004036325, 11.98805358814459, 12.45492582298562, 12.91371819942394, 13.23713177169034
