| L(s) = 1 | − 0.732·3-s + 2.73·5-s − 7-s − 2.46·9-s + 3.46·11-s − 1.46·13-s − 2·15-s − 0.732·17-s + 2·19-s + 0.732·21-s − 23-s + 2.46·25-s + 4·27-s + 8·29-s − 0.196·31-s − 2.53·33-s − 2.73·35-s + 0.535·37-s + 1.07·39-s − 2·41-s − 6.73·45-s + 8.19·47-s + 49-s + 0.535·51-s + 3.46·53-s + 9.46·55-s − 1.46·57-s + ⋯ |
| L(s) = 1 | − 0.422·3-s + 1.22·5-s − 0.377·7-s − 0.821·9-s + 1.04·11-s − 0.406·13-s − 0.516·15-s − 0.177·17-s + 0.458·19-s + 0.159·21-s − 0.208·23-s + 0.492·25-s + 0.769·27-s + 1.48·29-s − 0.0352·31-s − 0.441·33-s − 0.461·35-s + 0.0881·37-s + 0.171·39-s − 0.312·41-s − 1.00·45-s + 1.19·47-s + 0.142·49-s + 0.0750·51-s + 0.475·53-s + 1.27·55-s − 0.193·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834262113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834262113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 0.196T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 0.928T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
| 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
−8.975467280779057927428962009995, −8.331134593083110528843729670496, −7.07536060057912374564141171637, −6.47678576704012965980384156630, −5.81587198784447117779361922356, −5.21295420578910448486224744095, −4.15532164612506301777646392871, −3.00693721043437183463575815551, −2.14922611924740239663352525077, −0.885429537607292244517348090749,
0.885429537607292244517348090749, 2.14922611924740239663352525077, 3.00693721043437183463575815551, 4.15532164612506301777646392871, 5.21295420578910448486224744095, 5.81587198784447117779361922356, 6.47678576704012965980384156630, 7.07536060057912374564141171637, 8.331134593083110528843729670496, 8.975467280779057927428962009995
