L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.5 + 2.59i)3-s + (3.5 − 6.06i)4-s + (6 + 10.3i)5-s − 3·6-s + 15·8-s + (−4.5 − 7.79i)9-s + (−6 + 10.3i)10-s + (−10 + 17.3i)11-s + (10.5 + 18.1i)12-s + 84·13-s − 36·15-s + (−20.5 − 35.5i)16-s + (−48 + 83.1i)17-s + (4.5 − 7.79i)18-s + (6 + 10.3i)19-s + ⋯ |
L(s) = 1 | + (0.176 + 0.306i)2-s + (−0.288 + 0.499i)3-s + (0.437 − 0.757i)4-s + (0.536 + 0.929i)5-s − 0.204·6-s + 0.662·8-s + (−0.166 − 0.288i)9-s + (−0.189 + 0.328i)10-s + (−0.274 + 0.474i)11-s + (0.252 + 0.437i)12-s + 1.79·13-s − 0.619·15-s + (−0.320 − 0.554i)16-s + (−0.684 + 1.18i)17-s + (0.0589 − 0.102i)18-s + (0.0724 + 0.125i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.72255 + 1.14580i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72255 + 1.14580i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 84T + 2.19e3T^{2} \) |
| 17 | \( 1 + (48 - 83.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6 - 10.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-88 - 152. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 58T + 2.43e4T^{2} \) |
| 31 | \( 1 + (132 - 228. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (129 + 223. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 156T + 7.95e4T^{2} \) |
| 47 | \( 1 + (204 + 353. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-361 + 625. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-246 + 426. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (246 + 426. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (206 - 356. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 296T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-120 + 207. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 924T + 5.71e5T^{2} \) |
| 89 | \( 1 + (372 + 644. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 168T + 9.12e5T^{2} \) |
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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
−13.00315217691360439968708996263, −11.29978586278006022516808716364, −10.74495976163719189623829646560, −10.04608539458316671516943515909, −8.728429921162344833521069289658, −7.02712299851592954575599563764, −6.22196942783576288407042709100, −5.29723487834438971853423084639, −3.60792213211727936808986124303, −1.74107610204896069257049880972,
1.12351654957520832929954739098, 2.74989193822305464879217301569, 4.43073201132199910386145336512, 5.83747427087386114739726620448, 6.97592401978985199051807778662, 8.315399159451809975689180869226, 9.032229640691958750311850373664, 10.78925679984839434376170097909, 11.43039709778896198090859351612, 12.53737654406674568567178814914