| L(s) = 1 | + (−1.02 + 0.979i)2-s + (−1.63 + 1.11i)3-s + (0.0826 − 1.99i)4-s + (2.43 − 0.182i)5-s + (0.577 − 2.74i)6-s + (1.91 + 1.82i)7-s + (1.87 + 2.12i)8-s + (0.337 − 0.859i)9-s + (−2.30 + 2.57i)10-s + (−2.09 + 0.823i)11-s + (2.09 + 3.36i)12-s + (0.425 + 0.339i)13-s + (−3.74 + 0.00463i)14-s + (−3.78 + 3.01i)15-s + (−3.98 − 0.330i)16-s + (−2.31 + 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.721 + 0.692i)2-s + (−0.944 + 0.644i)3-s + (0.0413 − 0.999i)4-s + (1.08 − 0.0816i)5-s + (0.235 − 1.11i)6-s + (0.722 + 0.691i)7-s + (0.661 + 0.749i)8-s + (0.112 − 0.286i)9-s + (−0.729 + 0.813i)10-s + (−0.632 + 0.248i)11-s + (0.604 + 0.970i)12-s + (0.117 + 0.0940i)13-s + (−0.999 + 0.00123i)14-s + (−0.976 + 0.779i)15-s + (−0.996 − 0.0825i)16-s + (−0.562 + 0.606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377315 + 0.616698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377315 + 0.616698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 - 0.979i)T \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
| good | 3 | \( 1 + (1.63 - 1.11i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-2.43 + 0.182i)T + (4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (2.09 - 0.823i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.425 - 0.339i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (2.31 - 2.49i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.535i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.40 - 5.82i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.87 - 8.19i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.779 + 1.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-11.4 + 3.53i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (0.189 - 0.393i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (2.48 + 5.15i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.28 + 0.344i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.32 - 0.407i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.851 + 11.3i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (4.06 + 13.1i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (5.00 + 2.89i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.15 + 2.08i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.0617 - 0.409i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-5.72 + 3.30i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.68 - 7.12i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 0.879i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 16.6iT - 97T^{2} \) |
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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
−12.87349876005850017503366856722, −11.27853298618573098736480139493, −10.84151785131566691279195658692, −9.764013910537154760720441441503, −9.023257226873562151266832124272, −7.81681829774802727799471950698, −6.37110850401857206187662674471, −5.45284799855962929190333534901, −4.92097194907541784584189748568, −1.94539781430128658757605064105,
0.955856899306573965428602064407, 2.50963899366327486340994039708, 4.62331023946777626850552918624, 6.04477210281411513909401175764, 7.07404727994871619491724739312, 8.162180771656028572976941793899, 9.418188098331164778265459921714, 10.44504713951519309750733650479, 11.13257333211896226609910654397, 11.91035175257004534180725405793
