| L(s) = 1 | + 3.70·5-s − 7-s + 4.51·11-s + 13-s + 4.95·17-s − 3.27·19-s + 8.14·23-s + 8.76·25-s + 1.27·29-s + 10.2·31-s − 3.70·35-s − 4.95·37-s + 0.435·41-s − 8.14·43-s − 5.24·47-s + 49-s + 1.74·53-s + 16.7·55-s − 11.4·59-s − 2.56·61-s + 3.70·65-s − 12.2·67-s + 11.4·71-s + 14.1·73-s − 4.51·77-s − 3.67·79-s − 10.1·83-s + ⋯ |
| L(s) = 1 | + 1.65·5-s − 0.377·7-s + 1.36·11-s + 0.277·13-s + 1.20·17-s − 0.751·19-s + 1.69·23-s + 1.75·25-s + 0.236·29-s + 1.83·31-s − 0.627·35-s − 0.814·37-s + 0.0680·41-s − 1.24·43-s − 0.764·47-s + 0.142·49-s + 0.239·53-s + 2.25·55-s − 1.49·59-s − 0.328·61-s + 0.460·65-s − 1.50·67-s + 1.36·71-s + 1.65·73-s − 0.514·77-s − 0.413·79-s − 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.421058165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421058165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 17 | \( 1 - 4.95T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 - 8.14T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.95T + 37T^{2} \) |
| 41 | \( 1 - 0.435T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 - 1.74T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−8.176221226044261578414521460503, −6.94897720551359809013155546634, −6.54604094938992167796521877821, −6.02837936790002084673808881291, −5.24200838789580628420007805784, −4.53730567782966131173655109930, −3.40912013770260579253065967071, −2.78728019686838880609735016116, −1.65724978154692170193808555028, −1.07612740598429450872268836750,
1.07612740598429450872268836750, 1.65724978154692170193808555028, 2.78728019686838880609735016116, 3.40912013770260579253065967071, 4.53730567782966131173655109930, 5.24200838789580628420007805784, 6.02837936790002084673808881291, 6.54604094938992167796521877821, 6.94897720551359809013155546634, 8.176221226044261578414521460503
