L(s) = 1 | + (0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (0.500 − 0.866i)4-s − 5-s + (1.5 + 0.866i)6-s + (−1.5 + 2.59i)7-s + 3·8-s + (1.5 − 2.59i)9-s + (−0.5 − 0.866i)10-s + (−1.5 + 2.59i)11-s − 1.73i·12-s + (3 − 5.19i)13-s − 3·14-s + (−1.5 + 0.866i)15-s + (0.500 + 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s − 0.447·5-s + (0.612 + 0.353i)6-s + (−0.566 + 0.981i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s + (−0.158 − 0.273i)10-s + (−0.452 + 0.783i)11-s − 0.499i·12-s + (0.832 − 1.44i)13-s − 0.801·14-s + (−0.387 + 0.223i)15-s + (0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69941 + 0.130499i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69941 + 0.130499i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.01135616862345496556186435003, −12.14026648587108730965363885108, −10.70703578334391991843956196728, −9.662682133372304131530054979206, −8.402380373340848535792101374326, −7.60034681006466857306184100899, −6.43824348244610995544157926519, −5.48619817971487371812528013787, −3.72366236420306153594793405401, −2.11777052431038888001191696307,
2.38020460520154617673660805410, 3.86507867123677708209267961929, 4.20616057459367871776702764288, 6.56471602709823055667951663543, 7.74510585684968636519446495978, 8.598884274685412542893589621993, 9.920853486335495210077965541604, 10.86324642580606885432002952431, 11.59094825265085010171728403008, 12.97521832623334953220265396399