| L(s) = 1 | − 1.25·3-s − 2.93·5-s + 7-s − 1.42·9-s − 2·11-s + 3.36·13-s + 3.68·15-s − 17-s − 5.36·19-s − 1.25·21-s + 2.50·23-s + 3.61·25-s + 5.55·27-s − 4.37·29-s + 6.42·31-s + 2.50·33-s − 2.93·35-s + 1.01·37-s − 4.21·39-s + 8.48·41-s − 6.10·43-s + 4.18·45-s + 2.50·47-s + 49-s + 1.25·51-s + 11.3·53-s + 5.87·55-s + ⋯ |
| L(s) = 1 | − 0.724·3-s − 1.31·5-s + 0.377·7-s − 0.475·9-s − 0.603·11-s + 0.932·13-s + 0.950·15-s − 0.242·17-s − 1.23·19-s − 0.273·21-s + 0.522·23-s + 0.723·25-s + 1.06·27-s − 0.813·29-s + 1.15·31-s + 0.436·33-s − 0.496·35-s + 0.167·37-s − 0.675·39-s + 1.32·41-s − 0.931·43-s + 0.624·45-s + 0.365·47-s + 0.142·49-s + 0.175·51-s + 1.55·53-s + 0.791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 19 | \( 1 + 5.36T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 - 1.01T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 - 6.98T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.60490400878556510280709946629, −6.79573202830111803656852247913, −6.12919939572968740495190648217, −5.44347371595273215444274781698, −4.61722837516255419537215337869, −4.08483422244643669341178264896, −3.22866357463142974798718958764, −2.30191459103532578766357660923, −0.932711826185541892155116411525, 0,
0.932711826185541892155116411525, 2.30191459103532578766357660923, 3.22866357463142974798718958764, 4.08483422244643669341178264896, 4.61722837516255419537215337869, 5.44347371595273215444274781698, 6.12919939572968740495190648217, 6.79573202830111803656852247913, 7.60490400878556510280709946629
