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
−11.21810675608218680373791521519, −9.915290945990282275816981177626, −9.349200649359226397204232495835, −8.397349950357341371599555349589, −7.45068772738126612981702199985, −6.71592949044323375709232541164, −5.78057054792255926126464675400, −4.21892941842330540272959295355, −2.53648597551416542984628681990, −1.11426541522230063067801729219,
1.92019953678949994193098644438, 3.19514922540507771440334930714, 4.25109253051187980106023021469, 5.61885975297480890558703521814, 6.89624508564842379227055771840, 8.110490778637511514637556069190, 9.094186720611141465312649254595, 9.531513359134655701454717397999, 10.35939273529317271645955806289, 11.44238566026926547160821088845