| L(s) = 1 | + (1.14 − 0.823i)2-s + (−0.831 + 0.555i)3-s + (0.644 − 1.89i)4-s + (1.87 − 0.373i)5-s + (−0.498 + 1.32i)6-s + (−0.0424 − 0.102i)7-s + (−0.816 − 2.70i)8-s + (0.382 − 0.923i)9-s + (1.85 − 1.97i)10-s + (−0.404 + 0.605i)11-s + (0.515 + 1.93i)12-s + (2.24 + 0.446i)13-s + (−0.133 − 0.0828i)14-s + (−1.35 + 1.35i)15-s + (−3.16 − 2.44i)16-s + (0.165 + 0.165i)17-s + ⋯ |
| L(s) = 1 | + (0.813 − 0.582i)2-s + (−0.480 + 0.320i)3-s + (0.322 − 0.946i)4-s + (0.840 − 0.167i)5-s + (−0.203 + 0.540i)6-s + (−0.0160 − 0.0387i)7-s + (−0.288 − 0.957i)8-s + (0.127 − 0.307i)9-s + (0.586 − 0.625i)10-s + (−0.121 + 0.182i)11-s + (0.148 + 0.557i)12-s + (0.623 + 0.123i)13-s + (−0.0355 − 0.0221i)14-s + (−0.349 + 0.349i)15-s + (−0.792 − 0.610i)16-s + (0.0400 + 0.0400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55316 - 0.737695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55316 - 0.737695i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.14 + 0.823i)T \) |
| 3 | \( 1 + (0.831 - 0.555i)T \) |
| good | 5 | \( 1 + (-1.87 + 0.373i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (0.0424 + 0.102i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.404 - 0.605i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-2.24 - 0.446i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-0.165 - 0.165i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.641 - 3.22i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (4.98 + 2.06i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.724 - 1.08i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 - 9.25iT - 31T^{2} \) |
| 37 | \( 1 + (-2.04 - 10.2i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (7.84 + 3.24i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.570 + 0.381i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (2.61 + 2.61i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.35 + 3.53i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-12.7 + 2.52i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (2.44 - 1.63i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 1.39i)T + (25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (4.56 + 11.0i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.20 + 10.1i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (9.56 - 9.56i)T - 79iT^{2} \) |
| 83 | \( 1 + (-2.32 + 11.6i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (7.91 - 3.27i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 17.6iT - 97T^{2} \) |
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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
−12.34328650338185820254572541666, −11.58438150500840715017969915304, −10.32886218307333785306062980537, −9.999866691002875658869391599326, −8.615565927610293429183848465228, −6.71865610787992149456173447302, −5.83657768797627665024552263177, −4.87183284635208335607027365689, −3.54716464538877895192266537838, −1.75112691642764895671677445284,
2.35508033673380516360691374554, 4.09121227100828510253498288518, 5.58657510083440223707166585121, 6.12820503555755673916162842393, 7.27492426767557700071947423104, 8.387264626477496683766542814618, 9.738343534733267889978222556190, 11.02433471281185436719552235722, 11.82422591485099579076355125913, 12.99707080190589699436248652870
