L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s + 16-s + 8·17-s + 18-s − 19-s − 2·22-s + 6·23-s + 24-s + 27-s − 2·29-s + 8·31-s + 32-s − 2·33-s + 8·34-s + 36-s + 10·37-s − 38-s − 2·41-s + 8·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 0.229·19-s − 0.426·22-s + 1.25·23-s + 0.204·24-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s + 1.37·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s − 0.312·41-s + 1.21·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.475232479\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.475232479\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.43806074164788, −12.83717063509140, −12.68630615767657, −12.09082972124834, −11.60915777996938, −11.03440918523374, −10.53470813144772, −10.08646288752999, −9.609034014049731, −9.062474224354587, −8.493485185747510, −7.750235863916758, −7.689269541691369, −7.134213586401390, −6.303962485789949, −5.989544082088007, −5.326708067800156, −4.877557458431825, −4.297513924071351, −3.715372724442104, −3.047876495944415, −2.775896486139901, −2.133755265640510, −1.193129118254454, −0.7722816705614071,
0.7722816705614071, 1.193129118254454, 2.133755265640510, 2.775896486139901, 3.047876495944415, 3.715372724442104, 4.297513924071351, 4.877557458431825, 5.326708067800156, 5.989544082088007, 6.303962485789949, 7.134213586401390, 7.689269541691369, 7.750235863916758, 8.493485185747510, 9.062474224354587, 9.609034014049731, 10.08646288752999, 10.53470813144772, 11.03440918523374, 11.60915777996938, 12.09082972124834, 12.68630615767657, 12.83717063509140, 13.43806074164788