| L(s) = 1 | + 6.46·3-s + 22.1·5-s − 246.·7-s − 201.·9-s + 121·11-s + 495.·13-s + 142.·15-s + 84.6·17-s + 2.83e3·19-s − 1.59e3·21-s − 3.45e3·23-s − 2.63e3·25-s − 2.87e3·27-s − 4.59e3·29-s + 5.52e3·31-s + 781.·33-s − 5.45e3·35-s − 5.55e3·37-s + 3.20e3·39-s − 2.05e4·41-s + 4.69e3·43-s − 4.45e3·45-s − 1.09e4·47-s + 4.40e4·49-s + 547.·51-s + 1.33e3·53-s + 2.67e3·55-s + ⋯ |
| L(s) = 1 | + 0.414·3-s + 0.395·5-s − 1.90·7-s − 0.828·9-s + 0.301·11-s + 0.813·13-s + 0.164·15-s + 0.0710·17-s + 1.80·19-s − 0.789·21-s − 1.36·23-s − 0.843·25-s − 0.757·27-s − 1.01·29-s + 1.03·31-s + 0.124·33-s − 0.753·35-s − 0.667·37-s + 0.337·39-s − 1.91·41-s + 0.387·43-s − 0.327·45-s − 0.725·47-s + 2.62·49-s + 0.0294·51-s + 0.0653·53-s + 0.119·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.625908379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625908379\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 6.46T + 243T^{2} \) |
| 5 | \( 1 - 22.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 246.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 495.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 84.6T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.83e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.30e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.47e4T + 8.58e9T^{2} \) |
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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
−9.744557719340947110103431364614, −8.961900971704783088623901552830, −8.055214595563953741216061111780, −6.91951309314448458916471192252, −6.09401404798080714234353605052, −5.48961937291403889921831847875, −3.64236917900063615445330919696, −3.31552693550102817731089447932, −2.08624607986613895264076875895, −0.56162249255901091311369490251,
0.56162249255901091311369490251, 2.08624607986613895264076875895, 3.31552693550102817731089447932, 3.64236917900063615445330919696, 5.48961937291403889921831847875, 6.09401404798080714234353605052, 6.91951309314448458916471192252, 8.055214595563953741216061111780, 8.961900971704783088623901552830, 9.744557719340947110103431364614
