L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 3·11-s + 13-s − 3·15-s + 17-s + 4·19-s + 4·23-s + 25-s − 9·27-s + 9·29-s − 6·31-s − 9·33-s + 8·37-s − 3·39-s − 6·41-s + 8·43-s + 6·45-s + 7·47-s − 3·51-s + 8·53-s + 3·55-s − 12·57-s + 4·59-s + 10·61-s + 65-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.904·11-s + 0.277·13-s − 0.774·15-s + 0.242·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 1.07·31-s − 1.56·33-s + 1.31·37-s − 0.480·39-s − 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.02·47-s − 0.420·51-s + 1.09·53-s + 0.404·55-s − 1.58·57-s + 0.520·59-s + 1.28·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.800626634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.800626634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.24456183848685, −15.65953335370342, −14.95357848854886, −14.26758835949399, −13.77505012694321, −12.94638284275690, −12.64330307085660, −11.83038449638971, −11.63179172021720, −11.01233359512141, −10.40309173190206, −9.943783101243967, −9.272323330327034, −8.691027183603121, −7.709022955811525, −6.920461067556201, −6.704575706174788, −5.800882700277038, −5.588395368535054, −4.803531647470070, −4.204412590789519, −3.369595035390939, −2.284296148257007, −1.118396845796386, −0.8242952598730138,
0.8242952598730138, 1.118396845796386, 2.284296148257007, 3.369595035390939, 4.204412590789519, 4.803531647470070, 5.588395368535054, 5.800882700277038, 6.704575706174788, 6.920461067556201, 7.709022955811525, 8.691027183603121, 9.272323330327034, 9.943783101243967, 10.40309173190206, 11.01233359512141, 11.63179172021720, 11.83038449638971, 12.64330307085660, 12.94638284275690, 13.77505012694321, 14.26758835949399, 14.95357848854886, 15.65953335370342, 16.24456183848685