| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s + 6·11-s − 12-s + 2·13-s + 14-s + 15-s − 16-s − 7·17-s + 18-s + 5·19-s − 20-s + 21-s + 6·22-s − 4·23-s − 3·24-s + 25-s + 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 1.69·17-s + 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 113 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.94503712313104, −13.53786477560482, −13.29887718276909, −12.65996671254580, −12.11043402456818, −11.62067535270905, −11.23483030842557, −10.66114238949706, −9.722783597468413, −9.510983106477183, −9.090990614827263, −8.599360874042067, −8.232939460261726, −7.364160540537224, −6.829015369021956, −6.327067716482913, −5.883838064752676, −5.115804189102323, −4.763977976287251, −3.983069557340950, −3.661729625145434, −3.303090465851065, −2.179144223666138, −1.817799245552974, −1.070753633141193, 0,
1.070753633141193, 1.817799245552974, 2.179144223666138, 3.303090465851065, 3.661729625145434, 3.983069557340950, 4.763977976287251, 5.115804189102323, 5.883838064752676, 6.327067716482913, 6.829015369021956, 7.364160540537224, 8.232939460261726, 8.599360874042067, 9.090990614827263, 9.510983106477183, 9.722783597468413, 10.66114238949706, 11.23483030842557, 11.62067535270905, 12.11043402456818, 12.65996671254580, 13.29887718276909, 13.53786477560482, 13.94503712313104
