| L(s) = 1 | − 3-s − 5-s − 5·7-s − 2·9-s − 11-s + 2·13-s + 15-s + 3·17-s + 7·19-s + 5·21-s + 6·23-s + 25-s + 5·27-s − 3·29-s + 7·31-s + 33-s + 5·35-s − 7·37-s − 2·39-s + 6·41-s − 8·43-s + 2·45-s − 6·47-s + 18·49-s − 3·51-s − 3·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s + 1.60·19-s + 1.09·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 1.25·31-s + 0.174·33-s + 0.845·35-s − 1.15·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.875·47-s + 18/7·49-s − 0.420·51-s − 0.412·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7777156559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7777156559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−10.10601413049489588702141344635, −9.422908134669636162293514884732, −8.544737917210983194690740059198, −7.42866571243096715516314299317, −6.61479287984975987379606558452, −5.84669947312883121809728720280, −5.01647058733648301862383016754, −3.40544989879851656879800288228, −3.06066609883175896466733159370, −0.70791386076572374446358125906,
0.70791386076572374446358125906, 3.06066609883175896466733159370, 3.40544989879851656879800288228, 5.01647058733648301862383016754, 5.84669947312883121809728720280, 6.61479287984975987379606558452, 7.42866571243096715516314299317, 8.544737917210983194690740059198, 9.422908134669636162293514884732, 10.10601413049489588702141344635
