| L(s) = 1 | + 2.61·2-s − 1.15·4-s − 13.5·5-s − 7·7-s − 23.9·8-s − 35.4·10-s + 24.0·11-s + 23.9·13-s − 18.3·14-s − 53.3·16-s + 79.6·17-s − 50.0·19-s + 15.6·20-s + 62.9·22-s − 151.·23-s + 58.4·25-s + 62.7·26-s + 8.10·28-s + 256.·29-s − 2.72·31-s + 51.9·32-s + 208.·34-s + 94.8·35-s + 319.·37-s − 130.·38-s + 324.·40-s + 164.·41-s + ⋯ |
| L(s) = 1 | + 0.924·2-s − 0.144·4-s − 1.21·5-s − 0.377·7-s − 1.05·8-s − 1.12·10-s + 0.659·11-s + 0.511·13-s − 0.349·14-s − 0.834·16-s + 1.13·17-s − 0.604·19-s + 0.175·20-s + 0.610·22-s − 1.37·23-s + 0.467·25-s + 0.473·26-s + 0.0546·28-s + 1.64·29-s − 0.0157·31-s + 0.287·32-s + 1.05·34-s + 0.457·35-s + 1.42·37-s − 0.558·38-s + 1.28·40-s + 0.627·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.851395473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851395473\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 2.61T + 8T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 11 | \( 1 - 24.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.72T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 576.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 43.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 509.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 894.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 14.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.77e3T + 9.12e5T^{2} \) |
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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
−10.43948616120526233344756711558, −9.420848867755108582633476384082, −8.466941508277955479354656591010, −7.70614795085241671650515666384, −6.44947805028976175359000412147, −5.71687369840183267305561694010, −4.28300821146298440733708443733, −3.95510620220005947590686049068, −2.85376954703994484051617039885, −0.71526956154275712030976487626,
0.71526956154275712030976487626, 2.85376954703994484051617039885, 3.95510620220005947590686049068, 4.28300821146298440733708443733, 5.71687369840183267305561694010, 6.44947805028976175359000412147, 7.70614795085241671650515666384, 8.466941508277955479354656591010, 9.420848867755108582633476384082, 10.43948616120526233344756711558
