| L(s) = 1 | − 1.88·3-s − 1.20·5-s + 0.545·9-s + 11-s + 3.61·13-s + 2.27·15-s + 6.15·17-s − 6.05·19-s + 8.07·23-s − 3.53·25-s + 4.62·27-s + 8.85·29-s + 1.34·31-s − 1.88·33-s + 1.50·37-s − 6.79·39-s − 2.44·41-s − 6.85·43-s − 0.659·45-s + 4.74·47-s − 11.5·51-s + 14.2·53-s − 1.20·55-s + 11.4·57-s − 14.1·59-s − 7.97·61-s − 4.36·65-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 0.540·5-s + 0.181·9-s + 0.301·11-s + 1.00·13-s + 0.587·15-s + 1.49·17-s − 1.39·19-s + 1.68·23-s − 0.707·25-s + 0.889·27-s + 1.64·29-s + 0.242·31-s − 0.327·33-s + 0.247·37-s − 1.08·39-s − 0.382·41-s − 1.04·43-s − 0.0983·45-s + 0.691·47-s − 1.62·51-s + 1.95·53-s − 0.162·55-s + 1.51·57-s − 1.83·59-s − 1.02·61-s − 0.541·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233682430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233682430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.88T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 - 8.07T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 - 1.50T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 7.94T + 79T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 - 6.33T + 97T^{2} \) |
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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
−7.72399302307083659228508302721, −6.92434938137036030656886415913, −6.31099426185790514727383188571, −5.81066509661734800536232460574, −5.02172113335205037704189234991, −4.37906150313484722123562327196, −3.54237809583566115456065229726, −2.79614590370048544992326716338, −1.38947116773815344134116882242, −0.63498940801745792889827996688,
0.63498940801745792889827996688, 1.38947116773815344134116882242, 2.79614590370048544992326716338, 3.54237809583566115456065229726, 4.37906150313484722123562327196, 5.02172113335205037704189234991, 5.81066509661734800536232460574, 6.31099426185790514727383188571, 6.92434938137036030656886415913, 7.72399302307083659228508302721
