| L(s) = 1 | − 2.58·2-s + 3.61·3-s − 1.31·4-s + 3.92·5-s − 9.34·6-s + 24.0·8-s − 13.9·9-s − 10.1·10-s + 51.3·11-s − 4.74·12-s − 80.0·13-s + 14.1·15-s − 51.7·16-s + 12.9·17-s + 36.0·18-s − 124.·19-s − 5.14·20-s − 132.·22-s − 172.·23-s + 87.0·24-s − 109.·25-s + 207.·26-s − 147.·27-s − 151.·29-s − 36.6·30-s + 122.·31-s − 58.7·32-s + ⋯ |
| L(s) = 1 | − 0.914·2-s + 0.695·3-s − 0.164·4-s + 0.350·5-s − 0.635·6-s + 1.06·8-s − 0.516·9-s − 0.320·10-s + 1.40·11-s − 0.114·12-s − 1.70·13-s + 0.243·15-s − 0.808·16-s + 0.184·17-s + 0.472·18-s − 1.50·19-s − 0.0575·20-s − 1.28·22-s − 1.56·23-s + 0.739·24-s − 0.876·25-s + 1.56·26-s − 1.05·27-s − 0.970·29-s − 0.222·30-s + 0.711·31-s − 0.324·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9388566893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9388566893\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2.58T + 8T^{2} \) |
| 3 | \( 1 - 3.61T + 27T^{2} \) |
| 5 | \( 1 - 3.92T + 125T^{2} \) |
| 11 | \( 1 - 51.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 213.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 201.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 162.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 676.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 240.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 559.T + 9.12e5T^{2} \) |
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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
−8.832817136675355309164850957997, −7.911901230535104987502183176203, −7.48899779442938179397199674175, −6.40746942456934499319557984348, −5.60208507734984211902377504878, −4.36574506176664041919525513847, −3.87555610160992344012310114611, −2.38114019897628461545493817865, −1.88649275453226010700877458432, −0.46225091140864309699566194872,
0.46225091140864309699566194872, 1.88649275453226010700877458432, 2.38114019897628461545493817865, 3.87555610160992344012310114611, 4.36574506176664041919525513847, 5.60208507734984211902377504878, 6.40746942456934499319557984348, 7.48899779442938179397199674175, 7.911901230535104987502183176203, 8.832817136675355309164850957997
