| L(s) = 1 | + (0.909 + 1.08i)2-s + (−0.347 + 1.96i)4-s + (−2.86 − 7.86i)5-s + (−1.95 − 11.0i)7-s + (−2.44 + 1.41i)8-s + (5.92 − 10.2i)10-s + (−0.538 + 1.47i)11-s + (−6.82 − 5.72i)13-s + (10.2 − 12.1i)14-s + (−3.75 − 1.36i)16-s + (16.9 + 9.80i)17-s + (4.86 + 8.42i)19-s + (16.4 − 2.90i)20-s + (−2.09 + 0.761i)22-s + (13.8 + 2.43i)23-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.0868 + 0.492i)4-s + (−0.572 − 1.57i)5-s + (−0.278 − 1.58i)7-s + (−0.306 + 0.176i)8-s + (0.592 − 1.02i)10-s + (−0.0489 + 0.134i)11-s + (−0.524 − 0.440i)13-s + (0.729 − 0.869i)14-s + (−0.234 − 0.0855i)16-s + (0.999 + 0.576i)17-s + (0.256 + 0.443i)19-s + (0.824 − 0.145i)20-s + (−0.0950 + 0.0345i)22-s + (0.601 + 0.105i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.16654 - 0.784422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16654 - 0.784422i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.86 + 7.86i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.95 + 11.0i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (0.538 - 1.47i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (6.82 + 5.72i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-16.9 - 9.80i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4.86 - 8.42i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-13.8 - 2.43i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (7.05 + 8.40i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-8.04 + 45.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-7.89 + 13.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-6.59 + 7.86i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-24.6 - 8.96i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-42.2 + 7.45i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (20.6 + 56.8i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-13.2 - 75.3i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-2.09 - 1.75i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (65.1 + 37.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.0 - 57.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.8 + 32.5i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (21.4 + 25.5i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-75.2 + 43.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-132. - 48.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.72845326571931392778342737557, −11.77840078792474663965976025920, −10.36013836608834487374893232833, −9.265298512885877006476566758000, −7.906055906854416367334101961512, −7.46559076067287263046451698805, −5.77083296686908335568370861197, −4.57642944518190293500379854778, −3.74765930463654151707219078021, −0.78558857896081407356740298166,
2.55608652966409329282995061890, 3.29615560217415765965661079517, 5.13001408997074708600571394357, 6.34881208106725015155347268801, 7.39396952456832447857486522526, 8.927148523880973461460950922470, 9.983353255853872896992980750146, 11.05840921623778927704327136630, 11.83713387266071242216682732063, 12.48246200960769678229851261685
