L(s) = 1 | + (2.39 + 1.74i)3-s + (2.15 + 0.587i)5-s − 1.83·7-s + (1.79 + 5.51i)9-s + (0.566 − 1.74i)11-s + (−0.747 − 2.29i)13-s + (4.15 + 5.17i)15-s + (−2.25 + 1.63i)17-s + (−1.35 + 0.982i)19-s + (−4.39 − 3.19i)21-s + (2.39 − 7.38i)23-s + (4.30 + 2.53i)25-s + (−2.56 + 7.89i)27-s + (−6.13 − 4.45i)29-s + (−4.28 + 3.11i)31-s + ⋯ |
L(s) = 1 | + (1.38 + 1.00i)3-s + (0.964 + 0.262i)5-s − 0.692·7-s + (0.597 + 1.83i)9-s + (0.170 − 0.525i)11-s + (−0.207 − 0.637i)13-s + (1.07 + 1.33i)15-s + (−0.546 + 0.396i)17-s + (−0.310 + 0.225i)19-s + (−0.960 − 0.697i)21-s + (0.500 − 1.54i)23-s + (0.861 + 0.507i)25-s + (−0.493 + 1.52i)27-s + (−1.13 − 0.827i)29-s + (−0.769 + 0.558i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97643 + 1.04911i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97643 + 1.04911i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.15 - 0.587i)T \) |
good | 3 | \( 1 + (-2.39 - 1.74i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + (-0.566 + 1.74i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.747 + 2.29i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.25 - 1.63i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.35 - 0.982i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.39 + 7.38i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (6.13 + 4.45i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.28 - 3.11i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.406 + 1.25i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 3.34i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + (-1.48 - 1.07i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.27 - 3.83i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.79 - 8.61i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.799 + 2.46i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.68 + 5.58i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.247 - 0.179i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.61 + 14.2i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.79 + 2.03i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.15 - 3.74i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.02 - 3.15i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (8.97 + 6.51i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.86726843867801256053443712887, −10.38099915952357646209978533599, −9.458831247937779360643240396896, −8.954595987456312434171062858642, −8.017017902870314991986685616485, −6.68780437063565502913232734171, −5.58795615531793078695199630072, −4.25113173112155267258420097714, −3.17543145479068565871373215748, −2.30455551011585162489497384073,
1.64040639573228072635620942374, 2.54781719052810988305857681016, 3.81675721201009887317606995797, 5.47359169554131047566452158426, 6.81902757652566001662832356877, 7.19971851627842122857096147480, 8.515219402501792244791775588737, 9.421035939024335112010735717972, 9.582069503776676500969995415288, 11.20366369346543130038124967534