| L(s) = 1 | + (−0.195 + 0.232i)2-s + (−1.57 + 0.722i)3-s + (0.331 + 1.87i)4-s + (−2.07 − 0.825i)5-s + (0.139 − 0.507i)6-s + (−3.36 − 0.592i)7-s + (−1.02 − 0.594i)8-s + (1.95 − 2.27i)9-s + (0.598 − 0.322i)10-s + (−2.75 + 1.00i)11-s + (−1.87 − 2.71i)12-s + (3.53 + 4.21i)13-s + (0.795 − 0.667i)14-s + (3.86 − 0.201i)15-s + (−3.24 + 1.18i)16-s + (−1.83 + 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.138 + 0.164i)2-s + (−0.908 + 0.417i)3-s + (0.165 + 0.939i)4-s + (−0.929 − 0.369i)5-s + (0.0569 − 0.207i)6-s + (−1.27 − 0.224i)7-s + (−0.363 − 0.210i)8-s + (0.652 − 0.758i)9-s + (0.189 − 0.102i)10-s + (−0.831 + 0.302i)11-s + (−0.542 − 0.784i)12-s + (0.980 + 1.16i)13-s + (0.212 − 0.178i)14-s + (0.998 − 0.0519i)15-s + (−0.811 + 0.295i)16-s + (−0.445 + 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0310712 + 0.321344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0310712 + 0.321344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.57 - 0.722i)T \) |
| 5 | \( 1 + (2.07 + 0.825i)T \) |
| good | 2 | \( 1 + (0.195 - 0.232i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (3.36 + 0.592i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.75 - 1.00i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.53 - 4.21i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.83 - 1.05i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.20 - 3.81i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.56 + 0.804i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.308 - 0.259i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.573 + 3.24i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.52 + 2.03i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.73 - 3.13i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.0361 - 0.0992i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (11.3 + 2.00i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 + (-5.15 - 1.87i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 6.74i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.45 + 2.92i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.75 - 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.69 + 3.86i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.80 + 1.51i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.37 - 5.21i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.870 + 1.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.152 + 0.417i)T + (-74.3 + 62.3i)T^{2} \) |
| 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
−13.13913432858820331396417483811, −12.70584198016803423313624972730, −11.68424004084315765461544609116, −10.86084308477513163720676195281, −9.531316077629762754451885475187, −8.450934903809661559773389963055, −7.12830997722416258237945169915, −6.28105636784009475010312160172, −4.42675812764730088717783799756, −3.50758357422665521913799923988,
0.37180964300720121695164609481, 3.00556850078411408208296551833, 5.06814458054513394780555160685, 6.21867679793034329571265070411, 7.01562591858964887801111937538, 8.510437536573045639757031398143, 10.03691151877569271586530402365, 10.85221611020923254222883155288, 11.44907982960136973845338597466, 12.82673408351542737686236729165
