L(s) = 1 | + (13.6 + 8.35i)2-s − 46.7i·3-s + (116. + 227. i)4-s + 904.·5-s + (390. − 638. i)6-s + 888. i·7-s + (−313. + 4.08e3i)8-s − 2.18e3·9-s + (1.23e4 + 7.55e3i)10-s − 1.44e4i·11-s + (1.06e4 − 5.44e3i)12-s − 1.16e4·13-s + (−7.41e3 + 1.21e4i)14-s − 4.22e4i·15-s + (−3.83e4 + 5.31e4i)16-s − 1.28e5·17-s + ⋯ |
L(s) = 1 | + (0.852 + 0.521i)2-s − 0.577i·3-s + (0.455 + 0.890i)4-s + 1.44·5-s + (0.301 − 0.492i)6-s + 0.369i·7-s + (−0.0765 + 0.997i)8-s − 0.333·9-s + (1.23 + 0.755i)10-s − 0.987i·11-s + (0.514 − 0.262i)12-s − 0.406·13-s + (−0.193 + 0.315i)14-s − 0.835i·15-s + (−0.585 + 0.810i)16-s − 1.53·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.63097 + 0.633426i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.63097 + 0.633426i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-13.6 - 8.35i)T \) |
| 3 | \( 1 + 46.7iT \) |
good | 5 | \( 1 - 904.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 888. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.44e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.16e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.28e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.25e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.41e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.15e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 2.01e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.01e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.61e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.75e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 4.81e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.23e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.14e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 4.60e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.33e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.21e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.89e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.23e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 1.99e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.45e6T + 3.93e15T^{2} \) |
| 97 | \( 1 + 8.74e7T + 7.83e15T^{2} \) |
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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
−17.95969005248901994080600242231, −17.12521331578421232757569602748, −15.41043311668607877073888510025, −13.73565341430217107141294951610, −13.23235856553115165899059322779, −11.37301366314951607181608921096, −8.922081436105566884733014418443, −6.74656992719755193580773944616, −5.43283531158378359482501058637, −2.43135289177607163127032926336,
2.14170964409159608157572734762, 4.58110670046201228427021405920, 6.29347885965188173133037257070, 9.636839998993331534292908737004, 10.58430648197745025048841818200, 12.57418026455988057495594246976, 13.89890854273163580486540100559, 14.96753764522878578040816840152, 16.73707506637825076448069838137, 18.18971838411060869657096029271