L(s) = 1 | + 1.51i·2-s + (−3.66 + 2.11i)3-s + 1.71·4-s + (−4.51 + 2.60i)5-s + (−3.20 − 5.54i)6-s + (1.23 + 0.713i)7-s + 8.63i·8-s + (4.47 − 7.75i)9-s + (−3.93 − 6.81i)10-s + 19.5·11-s + (−6.29 + 3.63i)12-s + (4.99 − 8.65i)13-s + (−1.07 + 1.86i)14-s + (11.0 − 19.1i)15-s − 6.19·16-s + (−12.5 + 21.7i)17-s + ⋯ |
L(s) = 1 | + 0.755i·2-s + (−1.22 + 0.706i)3-s + 0.428·4-s + (−0.902 + 0.520i)5-s + (−0.533 − 0.924i)6-s + (0.176 + 0.101i)7-s + 1.07i·8-s + (0.497 − 0.861i)9-s + (−0.393 − 0.681i)10-s + 1.78·11-s + (−0.524 + 0.302i)12-s + (0.384 − 0.665i)13-s + (−0.0770 + 0.133i)14-s + (0.735 − 1.27i)15-s − 0.387·16-s + (−0.737 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.348661 + 0.722395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.348661 + 0.722395i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-35.8 + 23.7i)T \) |
good | 2 | \( 1 - 1.51iT - 4T^{2} \) |
| 3 | \( 1 + (3.66 - 2.11i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (4.51 - 2.60i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 0.713i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 19.5T + 121T^{2} \) |
| 13 | \( 1 + (-4.99 + 8.65i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (12.5 - 21.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-9.29 + 5.36i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.7 + 28.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-13.9 - 8.03i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-3.71 - 6.43i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-18.9 + 10.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.3T + 1.68e3T^{2} \) |
| 47 | \( 1 - 78.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (13.1 + 22.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 - 39.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + (27.5 + 15.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-61.2 - 106. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (110. + 63.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-14.9 - 8.61i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (18.2 - 31.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (42.8 + 74.2i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (11.1 - 6.45i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 113.T + 9.40e3T^{2} \) |
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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
−16.08746284561513207397480433355, −15.31484957537775186995130159200, −14.44235459499319176627529012538, −12.12214310543745323117885399215, −11.34788935578765739664934637628, −10.53527765525260965537607325356, −8.480752817064465838809047194163, −6.85501472447015163419118969080, −5.90782498744645331303729215571, −4.12874949726980317289595958301,
1.15910080510984721688554023920, 4.14792090992533252317070540006, 6.27705337983703985648105029504, 7.35166944108373304283887689308, 9.369870350411390151717125407716, 11.28077281682895372767763952180, 11.70548011876227645745936539125, 12.26766920136914284784895778975, 13.82484754075056981276583367450, 15.72276378948677614550716929004