L(s) = 1 | + (−3.04 + 4.21i)3-s − 9.33i·5-s − 36.3i·7-s + (−8.50 − 25.6i)9-s − 48.4·11-s + 25.8·13-s + (39.3 + 28.3i)15-s + 74.2i·17-s + 82.9i·19-s + (153. + 110. i)21-s − 179.·23-s + 37.8·25-s + (133. + 42.1i)27-s + 122. i·29-s + 64.1i·31-s + ⋯ |
L(s) = 1 | + (−0.585 + 0.810i)3-s − 0.834i·5-s − 1.96i·7-s + (−0.314 − 0.949i)9-s − 1.32·11-s + 0.552·13-s + (0.676 + 0.488i)15-s + 1.05i·17-s + 1.00i·19-s + (1.59 + 1.14i)21-s − 1.62·23-s + 0.303·25-s + (0.953 + 0.300i)27-s + 0.783i·29-s + 0.371i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1167327886\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1167327886\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.04 - 4.21i)T \) |
good | 5 | \( 1 + 9.33iT - 125T^{2} \) |
| 7 | \( 1 + 36.3iT - 343T^{2} \) |
| 11 | \( 1 + 48.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 82.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 122. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 64.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.01T + 5.06e4T^{2} \) |
| 41 | \( 1 + 325. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 321. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 95.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 185. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 23.9iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 669.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 229. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 321.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 131. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 136.T + 9.12e5T^{2} \) |
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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
−10.78290622859770435805741426629, −10.56607920133618822854242873413, −9.779987574887698737788779188289, −8.446433907020420679564972567060, −7.68671916920883992580301351397, −6.36549930263378692384156809179, −5.31652403603385660695901653025, −4.29461953610219556317928440039, −3.62597183792813013522797909406, −1.23843732757127906647734889516,
0.04508028557609707913166222319, 2.25041013611078701875236429103, 2.78805677027714618908319133812, 5.02278494583374609597752356649, 5.79645263256011100929162625464, 6.57980481639727956807501234051, 7.72837227660588916989952423883, 8.505569620976048160027454298023, 9.678658857150671213863435251482, 10.81298860259311586342579736366