L(s) = 1 | + (1.86 − 0.731i)2-s − 1.73i·3-s + (2.92 − 2.72i)4-s − 3.15i·5-s + (−1.26 − 3.22i)6-s − 3.05·7-s + (3.46 − 7.21i)8-s − 2.99·9-s + (−2.30 − 5.87i)10-s − 3.10·11-s + (−4.71 − 5.07i)12-s + (9.09 + 9.28i)13-s + (−5.69 + 2.23i)14-s − 5.46·15-s + (1.16 − 15.9i)16-s + 20.0·17-s + ⋯ |
L(s) = 1 | + (0.930 − 0.365i)2-s − 0.577i·3-s + (0.732 − 0.680i)4-s − 0.630i·5-s + (−0.211 − 0.537i)6-s − 0.436·7-s + (0.432 − 0.901i)8-s − 0.333·9-s + (−0.230 − 0.587i)10-s − 0.281·11-s + (−0.393 − 0.422i)12-s + (0.699 + 0.714i)13-s + (−0.406 + 0.159i)14-s − 0.364·15-s + (0.0728 − 0.997i)16-s + 1.18·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0469 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0469 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.64537 - 1.72447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64537 - 1.72447i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.86 + 0.731i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 13 | \( 1 + (-9.09 - 9.28i)T \) |
good | 5 | \( 1 + 3.15iT - 25T^{2} \) |
| 7 | \( 1 + 3.05T + 49T^{2} \) |
| 11 | \( 1 + 3.10T + 121T^{2} \) |
| 17 | \( 1 - 20.0T + 289T^{2} \) |
| 19 | \( 1 + 12.4T + 361T^{2} \) |
| 23 | \( 1 - 5.55iT - 529T^{2} \) |
| 29 | \( 1 - 20.2T + 841T^{2} \) |
| 31 | \( 1 + 7.89T + 961T^{2} \) |
| 37 | \( 1 - 43.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 34.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 6.30T + 2.80e3T^{2} \) |
| 59 | \( 1 - 64.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 77.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 130.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 98.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 70.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 27.5iT - 9.40e3T^{2} \) |
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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
−12.55584459965528929983416132770, −11.79229266024388863354255027591, −10.68963862460396038067815708996, −9.509346739632906783279837626106, −8.180274019344058489873251359573, −6.80387948458260553172579268418, −5.84962048885485006540371641899, −4.57430440466643018620069226526, −3.09630793580812945209353267317, −1.34708958263041820362363109656,
2.84261871055466602096902869008, 3.84922152508806433517092835929, 5.33928468914287418196204509525, 6.30319834903676185600679346839, 7.50752135069925510534262471670, 8.686283561468603321643104742089, 10.30358753599805156421454913671, 10.88145987997532892890758058085, 12.16257181623382109384489709744, 13.01842027338117980597760717763