L(s) = 1 | + (0.636 + 1.10i)2-s + (0.5 + 0.866i)3-s + (0.188 − 0.326i)4-s + 1.10·5-s + (−0.636 + 1.10i)6-s + (0.5 − 0.866i)7-s + 3.02·8-s + (−0.499 + 0.866i)9-s + (0.702 + 1.21i)10-s + (0.174 + 0.302i)11-s + 0.377·12-s + (−3.47 − 0.955i)13-s + 1.27·14-s + (0.551 + 0.955i)15-s + (1.55 + 2.68i)16-s + (−0.363 + 0.628i)17-s + ⋯ |
L(s) = 1 | + (0.450 + 0.780i)2-s + (0.288 + 0.499i)3-s + (0.0943 − 0.163i)4-s + 0.493·5-s + (−0.260 + 0.450i)6-s + (0.188 − 0.327i)7-s + 1.07·8-s + (−0.166 + 0.288i)9-s + (0.222 + 0.384i)10-s + (0.0525 + 0.0911i)11-s + 0.108·12-s + (−0.964 − 0.265i)13-s + 0.340·14-s + (0.142 + 0.246i)15-s + (0.387 + 0.671i)16-s + (−0.0880 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71259 + 0.988768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71259 + 0.988768i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (3.47 + 0.955i)T \) |
good | 2 | \( 1 + (-0.636 - 1.10i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 + (-0.174 - 0.302i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.363 - 0.628i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.15 - 1.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0375 + 0.0649i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.240 + 0.416i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + (-0.551 - 0.955i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.54 + 2.68i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.31 + 4.00i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 + (-1.96 + 3.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.77 - 4.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.06 + 7.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.76 + 8.25i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + (-5.98 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.05 - 8.74i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.18028181935897057202222427052, −10.84378230548680461330973341941, −10.19859433030528479946308840367, −9.284681114837419404278266080908, −7.940220817347427605980481855576, −7.10904625384817948049368263650, −5.90183285358460665847405354335, −5.06811462902136490464263870260, −3.94604048598894512256666093342, −2.07348242146880424158847942121,
1.87450293777002994741027986699, 2.80810832438042863005868921978, 4.23993664862428089946512540646, 5.52257422488013204477100648040, 6.92004775534024443588714569134, 7.76894267451525893037169687176, 8.970955603047209334858397799928, 9.963270308964125646430202017787, 11.08881145907612941286132281497, 11.86643902846197667771481075133