| L(s) = 1 | + 3.37·3-s + 5-s − 3.37·7-s + 8.37·9-s + 11-s + 2·13-s + 3.37·15-s + 1.37·17-s − 0.627·19-s − 11.3·21-s − 2.74·23-s + 25-s + 18.1·27-s + 1.37·29-s − 3.37·31-s + 3.37·33-s − 3.37·35-s + 9.37·37-s + 6.74·39-s − 11.4·41-s + 4·43-s + 8.37·45-s − 2.74·47-s + 4.37·49-s + 4.62·51-s − 4.11·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s − 1.27·7-s + 2.79·9-s + 0.301·11-s + 0.554·13-s + 0.870·15-s + 0.332·17-s − 0.144·19-s − 2.48·21-s − 0.572·23-s + 0.200·25-s + 3.48·27-s + 0.254·29-s − 0.605·31-s + 0.587·33-s − 0.570·35-s + 1.54·37-s + 1.07·39-s − 1.79·41-s + 0.609·43-s + 1.24·45-s − 0.400·47-s + 0.624·49-s + 0.648·51-s − 0.565·53-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
| \(L(1)\) |
\(\approx\) |
\(3.055431996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.055431996\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.850487950012927257926481345266, −9.273971700822136757062469376054, −8.621437105409727728538132941072, −7.74731770890256922204915804004, −6.85794322406348863803057283312, −6.03293434937577325166887497596, −4.38179429008167304117657808191, −3.46291824474165386622167249990, −2.80319074640273897047538451374, −1.59946110200437612641870625021,
1.59946110200437612641870625021, 2.80319074640273897047538451374, 3.46291824474165386622167249990, 4.38179429008167304117657808191, 6.03293434937577325166887497596, 6.85794322406348863803057283312, 7.74731770890256922204915804004, 8.621437105409727728538132941072, 9.273971700822136757062469376054, 9.850487950012927257926481345266
