| L(s) = 1 | + 1.41i·2-s + (−2.82 + 1.00i)3-s − 2.00·4-s + (0.857 − 0.494i)5-s + (−1.41 − 3.99i)6-s + (−6.76 + 1.81i)7-s − 2.82i·8-s + (6.99 − 5.65i)9-s + (0.699 + 1.21i)10-s + (−17.4 − 10.0i)11-s + (5.65 − 2.00i)12-s + (5.16 − 8.94i)13-s + (−2.56 − 9.56i)14-s + (−1.92 + 2.25i)15-s + 4.00·16-s + (−5.20 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.942 + 0.333i)3-s − 0.500·4-s + (0.171 − 0.0989i)5-s + (−0.235 − 0.666i)6-s + (−0.965 + 0.259i)7-s − 0.353i·8-s + (0.777 − 0.628i)9-s + (0.0699 + 0.121i)10-s + (−1.58 − 0.913i)11-s + (0.471 − 0.166i)12-s + (0.397 − 0.687i)13-s + (−0.183 − 0.682i)14-s + (−0.128 + 0.150i)15-s + 0.250·16-s + (−0.306 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0500863 - 0.0747747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0500863 - 0.0747747i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (2.82 - 1.00i)T \) |
| 7 | \( 1 + (6.76 - 1.81i)T \) |
| good | 5 | \( 1 + (-0.857 + 0.494i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (17.4 + 10.0i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.16 + 8.94i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (5.20 - 3.00i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.72 + 15.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (28.6 - 16.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (21.5 - 12.4i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 15.9T + 961T^{2} \) |
| 37 | \( 1 + (20.2 - 35.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (51.3 + 29.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (10.4 + 18.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 0.295iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-25.8 + 14.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 49.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 57.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (27.1 + 47.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + 144.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-43.8 + 25.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (119. + 68.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-52.1 - 90.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.14777595991724733444992970674, −11.80358177874964802924572706817, −10.57497536611616013196209517338, −9.781094368430008106215134666963, −8.493065078775837555124095203625, −7.13699253035614016346686420426, −5.84641098836751347881796155127, −5.31455493052183532206953837247, −3.44574454712915667518181468324, −0.06579981332981331760452014354,
2.20363509472428956100662020479, 4.17464581898850850682435047498, 5.57081952127690950949578429786, 6.76898280761388873252190558446, 8.036209415938089179668387253279, 9.937165862940604102948717209949, 10.19601230806439036965385691842, 11.47718589685474370782005872685, 12.45258371552327412964031005713, 13.10643787269859866200045271822
