L(s) = 1 | + (2.5 − 4.33i)5-s + (−1 − 1.73i)7-s + (15 + 25.9i)11-s + (2 − 3.46i)13-s − 90·17-s − 28·19-s + (60 − 103. i)23-s + (−12.5 − 21.6i)25-s + (105 + 181. i)29-s + (2 − 3.46i)31-s − 10·35-s + 200·37-s + (120 − 207. i)41-s + (68 + 117. i)43-s + (−60 − 103. i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (−0.0539 − 0.0935i)7-s + (0.411 + 0.712i)11-s + (0.0426 − 0.0739i)13-s − 1.28·17-s − 0.338·19-s + (0.543 − 0.942i)23-s + (−0.100 − 0.173i)25-s + (0.672 + 1.16i)29-s + (0.0115 − 0.0200i)31-s − 0.0482·35-s + 0.888·37-s + (0.457 − 0.791i)41-s + (0.241 + 0.417i)43-s + (−0.186 − 0.322i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.097660147\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.097660147\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
good | 7 | \( 1 + (1 + 1.73i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 90T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-60 + 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-105 - 181. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 200T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-120 + 207. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-68 - 117. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (60 + 103. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 30T + 1.48e5T^{2} \) |
| 59 | \( 1 + (225 - 389. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-83 - 143. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (454 - 786. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 250T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-458 - 793. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (570 + 987. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 420T + 7.04e5T^{2} \) |
| 97 | \( 1 + (769 + 1.33e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−8.883759943364357859218304927373, −8.446319392553720728857708284538, −7.20050436109476826106362353364, −6.69634879381948113470005802642, −5.73808956093631785789058529017, −4.68188998467233301478501893869, −4.17237863640150590573124367733, −2.80725673286772482099290753093, −1.85016471102593187683118742496, −0.65143256734211370247081735996,
0.74324754200861791421041994190, 2.05587660198820689938867939007, 2.99694672027115655748303861153, 4.01123908595908201391263307822, 4.90673438356624743480239432816, 6.10246965044580990215061873354, 6.43978269491275225685642895930, 7.49537581278123381593460022759, 8.303412213753716073593860681159, 9.179481513238180583526154089039