| L(s) = 1 | + 2-s + 4-s − 5-s − 3·7-s + 8-s − 10-s + 3·11-s + 6·13-s − 3·14-s + 16-s + 2·17-s − 8·19-s − 20-s + 3·22-s − 6·23-s + 25-s + 6·26-s − 3·28-s − 8·31-s + 32-s + 2·34-s + 3·35-s − 10·37-s − 8·38-s − 40-s + 2·41-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.353·8-s − 0.316·10-s + 0.904·11-s + 1.66·13-s − 0.801·14-s + 1/4·16-s + 0.485·17-s − 1.83·19-s − 0.223·20-s + 0.639·22-s − 1.25·23-s + 1/5·25-s + 1.17·26-s − 0.566·28-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.507·35-s − 1.64·37-s − 1.29·38-s − 0.158·40-s + 0.312·41-s + 0.452·44-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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.35039848521578, −12.94717129750208, −12.61207071109241, −12.24820378792807, −11.54676652326525, −11.22272348568725, −10.76279509768228, −10.18848734848957, −9.782589506736416, −9.083853292198259, −8.647411346583067, −8.235799594342467, −7.663946339946780, −6.798944434212327, −6.625581045161639, −6.189230300920230, −5.751772532332242, −5.070245689831482, −4.314944129350899, −3.824860121985367, −3.514222579676345, −3.214647867085297, −1.976370941108655, −1.867500966358242, −0.8013034874964928, 0,
0.8013034874964928, 1.867500966358242, 1.976370941108655, 3.214647867085297, 3.514222579676345, 3.824860121985367, 4.314944129350899, 5.070245689831482, 5.751772532332242, 6.189230300920230, 6.625581045161639, 6.798944434212327, 7.663946339946780, 8.235799594342467, 8.647411346583067, 9.083853292198259, 9.782589506736416, 10.18848734848957, 10.76279509768228, 11.22272348568725, 11.54676652326525, 12.24820378792807, 12.61207071109241, 12.94717129750208, 13.35039848521578
