L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.992 + 1.41i)3-s + (0.173 + 0.984i)4-s + (0.999 + 0.363i)5-s + (0.152 − 1.72i)6-s + (−0.173 + 0.984i)7-s + (0.500 − 0.866i)8-s + (−1.03 + 2.81i)9-s + (−0.531 − 0.921i)10-s + (4.14 − 1.50i)11-s + (−1.22 + 1.22i)12-s + (−2.35 + 1.97i)13-s + (0.766 − 0.642i)14-s + (0.475 + 1.78i)15-s + (−0.939 + 0.342i)16-s + (0.333 + 0.578i)17-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.572 + 0.819i)3-s + (0.0868 + 0.492i)4-s + (0.447 + 0.162i)5-s + (0.0621 − 0.704i)6-s + (−0.0656 + 0.372i)7-s + (0.176 − 0.306i)8-s + (−0.343 + 0.939i)9-s + (−0.168 − 0.291i)10-s + (1.24 − 0.454i)11-s + (−0.353 + 0.353i)12-s + (−0.652 + 0.547i)13-s + (0.204 − 0.171i)14-s + (0.122 + 0.459i)15-s + (−0.234 + 0.0855i)16-s + (0.0809 + 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19965 + 0.579105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19965 + 0.579105i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.992 - 1.41i)T \) |
| 7 | \( 1 + (0.173 - 0.984i)T \) |
good | 5 | \( 1 + (-0.999 - 0.363i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-4.14 + 1.50i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.35 - 1.97i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.333 - 0.578i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.367 + 0.637i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.18 - 6.73i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 1.20i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.0404 + 0.229i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.60 - 6.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.28 + 6.95i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.07 - 2.57i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.85 + 10.5i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + (7.20 + 2.62i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0669 + 0.379i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.54 + 4.65i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.205 - 0.355i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.24 + 9.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.60 + 7.21i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (9.01 + 7.56i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.612 - 1.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.0 - 5.12i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.44238566026926547160821088845, −10.35939273529317271645955806289, −9.531513359134655701454717397999, −9.094186720611141465312649254595, −8.110490778637511514637556069190, −6.89624508564842379227055771840, −5.61885975297480890558703521814, −4.25109253051187980106023021469, −3.19514922540507771440334930714, −1.92019953678949994193098644438,
1.11426541522230063067801729219, 2.53648597551416542984628681990, 4.21892941842330540272959295355, 5.78057054792255926126464675400, 6.71592949044323375709232541164, 7.45068772738126612981702199985, 8.397349950357341371599555349589, 9.349200649359226397204232495835, 9.915290945990282275816981177626, 11.21810675608218680373791521519