| L(s) = 1 | + 2.51·2-s + 27.4·3-s − 25.6·4-s + 25·5-s + 69.1·6-s + 151.·7-s − 145.·8-s + 511.·9-s + 62.9·10-s − 121·11-s − 704.·12-s + 608.·13-s + 382.·14-s + 686.·15-s + 456.·16-s + 578.·17-s + 1.28e3·18-s − 361·19-s − 641.·20-s + 4.17e3·21-s − 304.·22-s + 4.86e3·23-s − 3.98e3·24-s + 625·25-s + 1.53e3·26-s + 7.36e3·27-s − 3.90e3·28-s + ⋯ |
| L(s) = 1 | + 0.444·2-s + 1.76·3-s − 0.802·4-s + 0.447·5-s + 0.783·6-s + 1.17·7-s − 0.801·8-s + 2.10·9-s + 0.198·10-s − 0.301·11-s − 1.41·12-s + 0.999·13-s + 0.521·14-s + 0.787·15-s + 0.445·16-s + 0.485·17-s + 0.935·18-s − 0.229·19-s − 0.358·20-s + 2.06·21-s − 0.134·22-s + 1.91·23-s − 1.41·24-s + 0.200·25-s + 0.444·26-s + 1.94·27-s − 0.940·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.787734900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.787734900\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 2.51T + 32T^{2} \) |
| 3 | \( 1 - 27.4T + 243T^{2} \) |
| 7 | \( 1 - 151.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 608.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 578.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 938.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.93e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.48e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−8.948304958475819337690244481583, −8.489043892374988528539767447450, −7.88084871560646261363570841150, −6.83517554045340190933386042050, −5.43526284393825709446600849299, −4.74978473168209362330507003779, −3.73265432260718308552663489705, −3.08868054547954536243851339697, −1.93838005673619663931247845513, −1.06181689878795972809405982566,
1.06181689878795972809405982566, 1.93838005673619663931247845513, 3.08868054547954536243851339697, 3.73265432260718308552663489705, 4.74978473168209362330507003779, 5.43526284393825709446600849299, 6.83517554045340190933386042050, 7.88084871560646261363570841150, 8.489043892374988528539767447450, 8.948304958475819337690244481583
