| L(s) = 1 | + 1.41·2-s + (−2.99 − 0.213i)3-s + 2.00·4-s + (−6.01 + 3.47i)5-s + (−4.23 − 0.302i)6-s + (−6.97 + 0.535i)7-s + 2.82·8-s + (8.90 + 1.27i)9-s + (−8.50 + 4.90i)10-s + (−8.72 + 15.1i)11-s + (−5.98 − 0.427i)12-s + (−13.3 − 7.69i)13-s + (−9.87 + 0.757i)14-s + (18.7 − 9.10i)15-s + 4.00·16-s + (12.3 − 7.13i)17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.997 − 0.0712i)3-s + 0.500·4-s + (−1.20 + 0.694i)5-s + (−0.705 − 0.0503i)6-s + (−0.997 + 0.0764i)7-s + 0.353·8-s + (0.989 + 0.142i)9-s + (−0.850 + 0.490i)10-s + (−0.793 + 1.37i)11-s + (−0.498 − 0.0356i)12-s + (−1.02 − 0.592i)13-s + (−0.705 + 0.0540i)14-s + (1.24 − 0.606i)15-s + 0.250·16-s + (0.726 − 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.128494 + 0.462670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128494 + 0.462670i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (2.99 + 0.213i)T \) |
| 7 | \( 1 + (6.97 - 0.535i)T \) |
| good | 5 | \( 1 + (6.01 - 3.47i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.72 - 15.1i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (13.3 + 7.69i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-12.3 + 7.13i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 0.724i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.43 - 11.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (2.98 + 5.17i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 16.4iT - 961T^{2} \) |
| 37 | \( 1 + (0.732 - 1.26i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-10.5 - 6.11i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (40.1 + 69.5i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 83.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-6.51 - 11.2i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 67.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 39.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 93.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-17.2 + 9.94i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-7.01 + 4.05i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (43.7 + 25.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-80.8 + 46.6i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.21259634751487425803231861879, −12.31626484751993598161588490609, −11.89159503472030451835766939205, −10.57585631607320319703317101644, −9.895951150885076654558366282666, −7.38138636413034168710060249468, −7.21936761052058107348109158061, −5.62710634863445942469922508972, −4.44152926433612004213285227220, −2.97087868837180771583327498797,
0.28521425423230390329029117448, 3.42140181860057022296294044877, 4.65910627383327621926200566836, 5.77167683044815027573602274133, 6.99465826717820585109082333000, 8.188048927827037107157165041504, 9.820693307443311506491119123764, 10.99876064011082553252866494368, 11.87390179016769985911259601680, 12.59041376550335785542426871813
