L(s) = 1 | − 2.43·2-s + (0.5 − 0.866i)3-s + 3.92·4-s + (−0.613 + 1.06i)5-s + (−1.21 + 2.10i)6-s + (2.20 + 1.46i)7-s − 4.68·8-s + (−0.499 − 0.866i)9-s + (1.49 − 2.58i)10-s + (−1.74 + 3.02i)11-s + (1.96 − 3.39i)12-s + (−2.87 + 2.17i)13-s + (−5.35 − 3.57i)14-s + (0.613 + 1.06i)15-s + 3.55·16-s + 4.52·17-s + ⋯ |
L(s) = 1 | − 1.72·2-s + (0.288 − 0.499i)3-s + 1.96·4-s + (−0.274 + 0.475i)5-s + (−0.496 + 0.860i)6-s + (0.831 + 0.554i)7-s − 1.65·8-s + (−0.166 − 0.288i)9-s + (0.472 − 0.818i)10-s + (−0.526 + 0.911i)11-s + (0.566 − 0.981i)12-s + (−0.797 + 0.603i)13-s + (−1.43 − 0.954i)14-s + (0.158 + 0.274i)15-s + 0.888·16-s + 1.09·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.527085 + 0.259353i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.527085 + 0.259353i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.20 - 1.46i)T \) |
| 13 | \( 1 + (2.87 - 2.17i)T \) |
good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 + (0.613 - 1.06i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.74 - 3.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 + (0.677 + 1.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.673T + 23T^{2} \) |
| 29 | \( 1 + (-2.64 - 4.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.99 - 8.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + (-3.61 - 6.25i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.48 + 7.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.58 + 4.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.95 + 8.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 0.803T + 59T^{2} \) |
| 61 | \( 1 + (2.32 + 4.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.06 + 1.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.52 - 4.37i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.04 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.90 - 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + (3.59 - 6.22i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.87776628530705989907302511574, −10.84589415252368104764162250979, −10.02937812220695921927804732544, −9.063394291554167310091869305209, −8.199414220711841711373349583479, −7.41643473585712227553572805929, −6.77991862958062977688835197685, −5.04131157984697139788552231327, −2.77177600359856111131933405575, −1.62435351962758991586416749725,
0.798210055929101775812197716720, 2.66637623860664639029813389259, 4.44915285171663175447171157560, 5.88480430956597432800915992902, 7.71972809530840792438085671436, 7.88160730689074343388096970080, 8.815075639017241304636691903478, 9.913619792174287349351268931472, 10.47664224045660020412812770618, 11.33692881129329679887929832701