L(s) = 1 | + 2.22i·2-s − 2.93·4-s + (0.701 + 0.404i)5-s + (−2.44 − 1.00i)7-s − 2.08i·8-s + (−0.900 + 1.55i)10-s + (2.66 + 1.53i)11-s + (−3.01 + 1.97i)13-s + (2.24 − 5.43i)14-s − 1.23·16-s − 2.79·17-s + (−3.73 + 2.15i)19-s + (−2.06 − 1.19i)20-s + (−3.41 + 5.92i)22-s − 4.95·23-s + ⋯ |
L(s) = 1 | + 1.57i·2-s − 1.46·4-s + (0.313 + 0.181i)5-s + (−0.924 − 0.381i)7-s − 0.738i·8-s + (−0.284 + 0.492i)10-s + (0.803 + 0.463i)11-s + (−0.836 + 0.548i)13-s + (0.599 − 1.45i)14-s − 0.309·16-s − 0.678·17-s + (−0.856 + 0.494i)19-s + (−0.461 − 0.266i)20-s + (−0.729 + 1.26i)22-s − 1.03·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.329871 - 0.439954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.329871 - 0.439954i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.44 + 1.00i)T \) |
| 13 | \( 1 + (3.01 - 1.97i)T \) |
good | 2 | \( 1 - 2.22iT - 2T^{2} \) |
| 5 | \( 1 + (-0.701 - 0.404i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.66 - 1.53i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2.79T + 17T^{2} \) |
| 19 | \( 1 + (3.73 - 2.15i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.95T + 23T^{2} \) |
| 29 | \( 1 + (-2.84 - 4.93i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.93 + 1.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.72iT - 37T^{2} \) |
| 41 | \( 1 + (8.48 - 4.90i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.85 - 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 3.06i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.83 + 4.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.93iT - 59T^{2} \) |
| 61 | \( 1 + (-1.51 - 2.62i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.84 - 4.52i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.75 - 1.58i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.13 + 1.97i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.52iT - 83T^{2} \) |
| 89 | \( 1 - 3.32iT - 89T^{2} \) |
| 97 | \( 1 + (-11.0 - 6.36i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.48105983018393565997723412875, −9.705849401372641534384767858307, −9.037744690113965564601726777221, −8.093670457621379252067826955149, −7.13160879552923775806058336863, −6.52855589311405999981292784471, −6.03084052380651656679648235857, −4.70573352542406640230433246524, −3.96062453400462737596572928363, −2.23358870705329788024744009262,
0.25333972238892839996631815147, 1.89826343171287628928915826645, 2.85093625750291024080902320972, 3.78870426372495260500094194732, 4.80992403272321365333503882238, 6.06657862647870705083880145302, 6.88768490261522388938596262567, 8.435913955647121020212725258093, 9.091092948672731763197824649613, 9.938640740311489189928739947341