| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 3·11-s − 4·13-s + 2·14-s + 16-s + 3·17-s + 19-s − 20-s − 3·22-s + 6·23-s + 25-s + 4·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s − 3·34-s + 2·35-s − 8·37-s − 38-s + 40-s + 9·41-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 + 0.904·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.223·20-s − 0.639·22-s + 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.338·35-s − 1.31·37-s − 0.162·38-s + 0.158·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.57908111102009, −12.71655008299261, −12.41950751843647, −12.06463983033401, −11.70402998509796, −10.85219723205771, −10.74102454109446, −10.05549858177775, −9.561908628108435, −9.186582323768466, −8.891202878550162, −8.107251988997298, −7.726431856516933, −7.120092937329299, −6.809369481143580, −6.398529402737917, −5.576161834702736, −5.191243381353917, −4.495946088603070, −3.817399071913439, −3.341877898517549, −2.768972275740528, −2.228526654055936, −1.278438019146812, −0.8070180577359259, 0,
0.8070180577359259, 1.278438019146812, 2.228526654055936, 2.768972275740528, 3.341877898517549, 3.817399071913439, 4.495946088603070, 5.191243381353917, 5.576161834702736, 6.398529402737917, 6.809369481143580, 7.120092937329299, 7.726431856516933, 8.107251988997298, 8.891202878550162, 9.186582323768466, 9.561908628108435, 10.05549858177775, 10.74102454109446, 10.85219723205771, 11.70402998509796, 12.06463983033401, 12.41950751843647, 12.71655008299261, 13.57908111102009
