L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (4.44 + 2.56i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−4.83 + 2.78i)11-s − 0.999·12-s + (−3.19 + 1.67i)13-s + 5.13·14-s + (−0.5 − 0.866i)16-s + (0.641 − 1.11i)17-s + 0.999i·18-s + (5.75 + 3.32i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (1.67 + 0.969i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−1.45 + 0.841i)11-s − 0.288·12-s + (−0.884 + 0.465i)13-s + 1.37·14-s + (−0.125 − 0.216i)16-s + (0.155 − 0.269i)17-s + 0.235i·18-s + (1.32 + 0.762i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.258008391\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.258008391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.19 - 1.67i)T \) |
good | 7 | \( 1 + (-4.44 - 2.56i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.83 - 2.78i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.641 + 1.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.75 - 3.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.43 - 5.94i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 + 3.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.30iT - 31T^{2} \) |
| 37 | \( 1 + (7.25 - 4.19i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 0.641i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 - 3.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 5.02iT - 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 + (-1.31 - 0.759i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.19 + 8.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.48 - 1.43i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.44 - 5.45i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 17.3iT - 83T^{2} \) |
| 89 | \( 1 + (-5.57 + 3.22i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.20 - 3.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.360940623482373799630019226367, −8.216362253389972843322571811420, −7.63679749031491530858610021783, −7.02066758528997558185432443105, −5.60928336132631228219470754762, −5.12654975575070019734959718714, −4.79903721177995175232894986350, −3.18100281593418208484812825592, −2.16982104585830832287519464383, −1.53475882109935599478377733146,
0.68725479704039656121289944768, 2.38814620319402301180362807248, 3.40884053276860434977443608915, 4.48548256202541032817168277765, 5.19962776803725060881755257325, 5.40961057529046285217950816713, 6.83881793318565847848035581838, 7.68006872436090494219121160835, 8.005725988950455104806513223309, 8.984746161213720149977348849201