L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 5·7-s + 0.999·8-s + (0.499 + 0.866i)10-s − 11-s + (−3 − 5.19i)13-s + (2.5 − 4.33i)14-s + (−0.5 + 0.866i)16-s + (2 − 3.46i)17-s + (−0.5 + 4.33i)19-s − 0.999·20-s + (0.5 − 0.866i)22-s + (3.5 + 6.06i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 1.88·7-s + 0.353·8-s + (0.158 + 0.273i)10-s − 0.301·11-s + (−0.832 − 1.44i)13-s + (0.668 − 1.15i)14-s + (−0.125 + 0.216i)16-s + (0.485 − 0.840i)17-s + (−0.114 + 0.993i)19-s − 0.223·20-s + (0.106 − 0.184i)22-s + (0.729 + 1.26i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6825048764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6825048764\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 4.33i)T \) |
good | 7 | \( 1 + 5T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.662880775712459049022105377144, −8.844054411717622077399636632895, −7.79775482413170400776768667274, −7.26672082479702954143164479571, −6.30450335167263285415955961060, −5.62887984335819565507149161373, −4.92603064421680306845631958872, −3.44533408316776937831906264339, −2.79838686637030373939818071606, −0.921348307351019644856337763920,
0.36994564203229181857154935077, 2.22309201653286083952235233638, 2.89382931247378853677318609190, 3.85939317970140680894921823197, 4.81316911907771002250255011138, 6.21472855405825764037049207558, 6.67138538879137382532202686114, 7.43366043197413088061239649776, 8.674753056178696405250714619088, 9.266738264866538870634634773207