| L(s) = 1 | + (1.41 − i)3-s − 2.82·5-s − 2.82i·7-s + (1.00 − 2.82i)9-s + 2i·11-s − 4i·13-s + (−4.00 + 2.82i)15-s − 5.65i·17-s − 2.82·19-s + (−2.82 − 4.00i)21-s + 8·23-s + 3.00·25-s + (−1.41 − 5.00i)27-s − 2.82·29-s + 8.48i·31-s + ⋯ |
| L(s) = 1 | + (0.816 − 0.577i)3-s − 1.26·5-s − 1.06i·7-s + (0.333 − 0.942i)9-s + 0.603i·11-s − 1.10i·13-s + (−1.03 + 0.730i)15-s − 1.37i·17-s − 0.648·19-s + (−0.617 − 0.872i)21-s + 1.66·23-s + 0.600·25-s + (−0.272 − 0.962i)27-s − 0.525·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.845803 - 1.00327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845803 - 1.00327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−11.12113728800711412455493601723, −10.21062315566546968952765229363, −9.058413775487829890219043506467, −8.120392734803400601703102035981, −7.30018491953753815298702269467, −6.93173108089170759841363258284, −4.94690964170703788921002533785, −3.83511327565108663264206937062, −2.89369658756497020588932642867, −0.832003487129236713181405303910,
2.27558610678440678205869454755, 3.61995776869795313834644009391, 4.35263455807574320277553178882, 5.71176362597283345245879739875, 7.08832167594432746985729948718, 8.207351237007476044536365432384, 8.710915475362624654605714114575, 9.502071417153748187244888192311, 10.90471790513756479044160850565, 11.38334284340090712234679305984
