L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.82 − 3.15i)5-s − 2.64·7-s − 0.999·8-s + (1.82 − 3.15i)10-s + (1.82 − 3.15i)11-s − 4.64·13-s + (−1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (1.82 − 3.15i)17-s + (−1 − 1.73i)19-s + 3.64·20-s + 3.64·22-s + (−0.645 − 1.11i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.815 − 1.41i)5-s − 0.999·7-s − 0.353·8-s + (0.576 − 0.998i)10-s + (0.549 − 0.951i)11-s − 1.28·13-s + (−0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (0.442 − 0.765i)17-s + (−0.229 − 0.397i)19-s + 0.815·20-s + 0.777·22-s + (−0.134 − 0.233i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.523816 - 0.558111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.523816 - 0.558111i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + (1.82 + 3.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.82 + 3.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 + 3.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.645 + 1.11i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 + (-2.32 + 4.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.96 - 10.3i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-2.46 - 4.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.17 + 7.23i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.14 + 1.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + (5.29 - 9.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.61 + 9.72i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (-2.46 - 4.27i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−11.60665478276366134308843303740, −9.883691839286054359074203856205, −9.133176837467216471041038355276, −8.296726630189243408480826634837, −7.38541746279037188231615354084, −6.29555167016919467141217585247, −5.15945834796982949152461143145, −4.30397081887324486776967804390, −3.11318357093167785124822727914, −0.44371697773959580542864949800,
2.35212087482632789877335464207, 3.43248451364571585966067193708, 4.26986174262211638135468398266, 5.89228674916507380439529391223, 6.95644672595677303560922918027, 7.54679335060942791678207626479, 9.144259109859848242965478774415, 10.18945282798098072819047880503, 10.49251565193596038216178532337, 11.85813281806120412180974253395