| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 5·11-s − 2·13-s + 14-s + 16-s + 17-s + 6·19-s − 5·22-s + 2·26-s − 28-s − 9·29-s + 3·31-s − 32-s − 34-s + 37-s − 6·38-s − 9·41-s − 43-s + 5·44-s − 6·49-s − 2·52-s + 9·53-s + 56-s + 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.50·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 1.06·22-s + 0.392·26-s − 0.188·28-s − 1.67·29-s + 0.538·31-s − 0.176·32-s − 0.171·34-s + 0.164·37-s − 0.973·38-s − 1.40·41-s − 0.152·43-s + 0.753·44-s − 6/7·49-s − 0.277·52-s + 1.23·53-s + 0.133·56-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.50451633947533, −15.72437343397659, −15.03083541571206, −14.69725873504937, −14.04836301113877, −13.43351764498027, −12.82031225840006, −12.03853326248525, −11.62452648830144, −11.39407913574793, −10.20047463482428, −10.07748512770582, −9.230164902607305, −9.060177731590933, −8.247136937264176, −7.414663058988314, −7.144981716112051, −6.410812913937196, −5.779373775285363, −5.095442620950930, −4.144399004132306, −3.477518266012799, −2.832460370222706, −1.759481137958505, −1.150289526514200, 0,
1.150289526514200, 1.759481137958505, 2.832460370222706, 3.477518266012799, 4.144399004132306, 5.095442620950930, 5.779373775285363, 6.410812913937196, 7.144981716112051, 7.414663058988314, 8.247136937264176, 9.060177731590933, 9.230164902607305, 10.07748512770582, 10.20047463482428, 11.39407913574793, 11.62452648830144, 12.03853326248525, 12.82031225840006, 13.43351764498027, 14.04836301113877, 14.69725873504937, 15.03083541571206, 15.72437343397659, 16.50451633947533
