| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s + 11-s − 2·14-s + 16-s − 2·17-s + 2·19-s + 20-s + 22-s + 2·23-s + 25-s − 2·28-s − 6·29-s − 8·31-s + 32-s − 2·34-s − 2·35-s − 4·37-s + 2·38-s + 40-s − 6·41-s + 44-s + 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s − 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.338·35-s − 0.657·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.150·44-s + 0.294·46-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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.38418318848576, −13.10151714069483, −12.59346142963064, −12.24343830561967, −11.48824108414824, −11.27539928111987, −10.69738496585439, −10.13548380753307, −9.687465315776420, −9.228935971692943, −8.782903965280454, −8.168052358273222, −7.454771925255716, −7.014388944761002, −6.630577114320305, −6.124254129050464, −5.498491853564711, −5.188338193014541, −4.592095680263396, −3.755581656064057, −3.519056622707745, −2.976970488501989, −2.073491674969236, −1.855154009164683, −0.8848283469183790, 0,
0.8848283469183790, 1.855154009164683, 2.073491674969236, 2.976970488501989, 3.519056622707745, 3.755581656064057, 4.592095680263396, 5.188338193014541, 5.498491853564711, 6.124254129050464, 6.630577114320305, 7.014388944761002, 7.454771925255716, 8.168052358273222, 8.782903965280454, 9.228935971692943, 9.687465315776420, 10.13548380753307, 10.69738496585439, 11.27539928111987, 11.48824108414824, 12.24343830561967, 12.59346142963064, 13.10151714069483, 13.38418318848576
