L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 1.73i·6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (1 − 1.73i)11-s + (1.49 + 0.866i)12-s + (3 + 5.19i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s + (1.5 + 2.59i)18-s + 6·19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.707i·6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.301 − 0.522i)11-s + (0.433 + 0.250i)12-s + (0.832 + 1.44i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.485·17-s + (0.353 + 0.612i)18-s + 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.688627 + 0.577826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.688627 + 0.577826i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-5.5 - 9.52i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.5 + 9.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.25219571113539281945455149920, −10.42720379826972787954285094758, −9.334755061906052980727749152159, −8.827790767311597196077133577803, −7.40394262571530211538012265832, −6.59864655247717234300780869072, −5.75115432739394494882569238313, −4.68192824884151139894029836659, −3.69376433286278946803766917838, −1.22545671346818461640141781313,
0.894228300865341295673256873882, 2.36285156628891161669948659412, 3.92029443215693296252491765638, 5.26664752326488959808064646856, 6.06685545682879593289550242577, 7.40462924476539799843451072119, 8.033183508616444963954119517298, 9.273141675490761525203275316781, 10.16431134189938566569845765730, 11.04726805948590608931520981806