| L(s) = 1 | + (−0.772 + 1.84i)2-s + (1.07 − 4.02i)3-s + (−2.80 − 2.84i)4-s + (−4.88 + 1.05i)5-s + (6.58 + 5.09i)6-s + (−4.65 + 5.22i)7-s + (7.42 − 2.98i)8-s + (−7.21 − 4.16i)9-s + (1.81 − 9.83i)10-s + (−8.45 + 4.88i)11-s + (−14.4 + 8.22i)12-s + (−11.7 + 11.7i)13-s + (−6.04 − 12.6i)14-s + (−1.00 + 20.7i)15-s + (−0.233 + 15.9i)16-s + (0.767 − 2.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.386 + 0.922i)2-s + (0.359 − 1.34i)3-s + (−0.701 − 0.712i)4-s + (−0.977 + 0.211i)5-s + (1.09 + 0.848i)6-s + (−0.665 + 0.746i)7-s + (0.928 − 0.372i)8-s + (−0.802 − 0.463i)9-s + (0.181 − 0.983i)10-s + (−0.769 + 0.444i)11-s + (−1.20 + 0.685i)12-s + (−0.902 + 0.902i)13-s + (−0.431 − 0.901i)14-s + (−0.0669 + 1.38i)15-s + (−0.0145 + 0.999i)16-s + (0.0451 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00146091 - 0.0272751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00146091 - 0.0272751i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.772 - 1.84i)T \) |
| 5 | \( 1 + (4.88 - 1.05i)T \) |
| 7 | \( 1 + (4.65 - 5.22i)T \) |
| good | 3 | \( 1 + (-1.07 + 4.02i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (8.45 - 4.88i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (11.7 - 11.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.767 + 2.86i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (19.1 + 11.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.05 + 15.1i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 15.6iT - 841T^{2} \) |
| 31 | \( 1 + (11.7 + 20.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-17.9 - 67.1i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 65.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (50.2 - 50.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-1.85 - 6.91i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-0.672 + 2.51i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (24.7 - 14.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.80 - 3.35i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (16.2 - 60.7i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 38.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (16.5 + 4.42i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-7.00 + 12.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (24.0 + 24.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-62.3 + 107. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-83.6 - 83.6i)T + 9.40e3iT^{2} \) |
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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
−13.37318432441413176156702487743, −12.65620424170530774795835672146, −11.70569253847911553131661206257, −10.10473736190957502329954656916, −8.837897616062321357248323958150, −7.960332932397576276867696950309, −7.09150475910714683517433081459, −6.37465591574068552066103661526, −4.63715238407043637254263103612, −2.40792932190637716005998219104,
0.01874421641476203878103676295, 3.13561671676310187678813517372, 3.90763347060372717666532375209, 5.03331993246002881699320254141, 7.51023446504274496405577144478, 8.439089334740421401870804047433, 9.548997576500146952393437578933, 10.43953952597039236431330026516, 10.91053471539861874970953250741, 12.36302804580789658150280551042
