| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s + 4·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 2·22-s − 6·23-s + 24-s + 25-s − 4·26-s − 27-s + 28-s + 6·29-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 − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.90307722491801, −13.37431482274878, −12.90662118652067, −12.29044255944733, −11.83906967846240, −11.46058677971835, −10.74081784162253, −10.56617565684747, −10.01273218106002, −9.642284275021572, −8.862925545619899, −8.456661585952045, −8.062617008595254, −7.448096175106176, −6.903051237422671, −6.292455338078290, −5.872605044368580, −5.491403630874408, −4.702252240208072, −4.250388137687463, −3.398839462677633, −2.853642161303794, −2.010303676022876, −1.533192460572291, −0.8540721697947956, 0,
0.8540721697947956, 1.533192460572291, 2.010303676022876, 2.853642161303794, 3.398839462677633, 4.250388137687463, 4.702252240208072, 5.491403630874408, 5.872605044368580, 6.292455338078290, 6.903051237422671, 7.448096175106176, 8.062617008595254, 8.456661585952045, 8.862925545619899, 9.642284275021572, 10.01273218106002, 10.56617565684747, 10.74081784162253, 11.46058677971835, 11.83906967846240, 12.29044255944733, 12.90662118652067, 13.37431482274878, 13.90307722491801
