| L(s) = 1 | − 0.517·2-s − 1.73·4-s − 2.44·7-s + 1.93·8-s − 11-s + 4.24·13-s + 1.26·14-s + 2.46·16-s + 0.378·17-s − 2·19-s + 0.517·22-s − 7.72·23-s − 2.19·26-s + 4.24·28-s + 2.53·29-s − 0.535·31-s − 5.13·32-s − 0.196·34-s − 4.89·37-s + 1.03·38-s + 2.53·41-s − 10.9·43-s + 1.73·44-s + 3.99·46-s + 2.82·47-s − 1.00·49-s − 7.34·52-s + ⋯ |
| L(s) = 1 | − 0.366·2-s − 0.866·4-s − 0.925·7-s + 0.683·8-s − 0.301·11-s + 1.17·13-s + 0.338·14-s + 0.616·16-s + 0.0919·17-s − 0.458·19-s + 0.110·22-s − 1.61·23-s − 0.430·26-s + 0.801·28-s + 0.470·29-s − 0.0962·31-s − 0.908·32-s − 0.0336·34-s − 0.805·37-s + 0.167·38-s + 0.396·41-s − 1.66·43-s + 0.261·44-s + 0.589·46-s + 0.412·47-s − 0.142·49-s − 1.01·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8279774620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8279774620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 0.378T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 7.72T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 + 0.535T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 0.656T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.744072839323906057410924109813, −8.449103197580767765866963790658, −7.56350818509711532409075790356, −6.54830328827603343784121939500, −5.89970154666442475866712664427, −4.99025350767165704306472586260, −3.93924248431285223443488299745, −3.43216139333094944743751971899, −2.00205261363848469319925502520, −0.60932291934523407899933154577,
0.60932291934523407899933154577, 2.00205261363848469319925502520, 3.43216139333094944743751971899, 3.93924248431285223443488299745, 4.99025350767165704306472586260, 5.89970154666442475866712664427, 6.54830328827603343784121939500, 7.56350818509711532409075790356, 8.449103197580767765866963790658, 8.744072839323906057410924109813
