| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s + 2·13-s + 16-s + 17-s − 4·19-s − 2·20-s − 4·22-s − 25-s − 2·26-s + 10·29-s − 8·31-s − 32-s − 34-s − 2·37-s + 4·38-s + 2·40-s + 10·41-s + 12·43-s + 4·44-s + 50-s + 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s + 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s − 0.328·37-s + 0.648·38-s + 0.316·40-s + 1.56·41-s + 1.82·43-s + 0.603·44-s + 0.141·50-s + 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291568782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291568782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.08671958474136, −15.74231339276110, −14.99639811746102, −14.35840334473615, −14.18232248030043, −13.04032861389880, −12.63687656278707, −11.87520154769727, −11.62480376494897, −10.88415306443631, −10.50946377457065, −9.695259301528106, −9.021757652480049, −8.670793040731763, −8.013009022590369, −7.409793600758856, −6.817333775098173, −6.185249793473940, −5.585450237447843, −4.353159195692036, −4.067557473920330, −3.265884145772982, −2.364033861082353, −1.392236561712433, −0.5987857765681408,
0.5987857765681408, 1.392236561712433, 2.364033861082353, 3.265884145772982, 4.067557473920330, 4.353159195692036, 5.585450237447843, 6.185249793473940, 6.817333775098173, 7.409793600758856, 8.013009022590369, 8.670793040731763, 9.021757652480049, 9.695259301528106, 10.50946377457065, 10.88415306443631, 11.62480376494897, 11.87520154769727, 12.63687656278707, 13.04032861389880, 14.18232248030043, 14.35840334473615, 14.99639811746102, 15.74231339276110, 16.08671958474136
