L(s) = 1 | + (1.06 + 0.926i)2-s + (−0.316 + 1.70i)3-s + (0.283 + 1.97i)4-s − 1.45i·5-s + (−1.91 + 1.52i)6-s + (1.17 + 0.680i)7-s + (−1.53 + 2.37i)8-s + (−2.80 − 1.07i)9-s + (1.34 − 1.55i)10-s + (−0.711 − 1.23i)11-s + (−3.46 − 0.142i)12-s + (3.27 − 1.50i)13-s + (0.629 + 1.82i)14-s + (2.47 + 0.458i)15-s + (−3.83 + 1.12i)16-s + (−1.68 − 0.971i)17-s + ⋯ |
L(s) = 1 | + (0.755 + 0.655i)2-s + (−0.182 + 0.983i)3-s + (0.141 + 0.989i)4-s − 0.648i·5-s + (−0.781 + 0.623i)6-s + (0.445 + 0.257i)7-s + (−0.541 + 0.840i)8-s + (−0.933 − 0.358i)9-s + (0.425 − 0.490i)10-s + (−0.214 − 0.371i)11-s + (−0.999 − 0.0412i)12-s + (0.908 − 0.417i)13-s + (0.168 + 0.486i)14-s + (0.638 + 0.118i)15-s + (−0.959 + 0.280i)16-s + (−0.408 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01828 + 1.16517i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01828 + 1.16517i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.06 - 0.926i)T \) |
| 3 | \( 1 + (0.316 - 1.70i)T \) |
| 13 | \( 1 + (-3.27 + 1.50i)T \) |
good | 5 | \( 1 + 1.45iT - 5T^{2} \) |
| 7 | \( 1 + (-1.17 - 0.680i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.711 + 1.23i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.68 + 0.971i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.01 - 1.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.96 + 3.39i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.48 + 4.32i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.55iT - 31T^{2} \) |
| 37 | \( 1 + (0.577 + 1.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.52 - 2.03i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (10.0 + 5.80i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 9.80iT - 53T^{2} \) |
| 59 | \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.92 - 8.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 6.32i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.89 - 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.21iT - 79T^{2} \) |
| 83 | \( 1 - 0.869T + 83T^{2} \) |
| 89 | \( 1 + (4.70 - 2.71i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.22 - 9.05i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.40402109697529861358795804463, −12.21956221910063964851163906267, −11.41462902711005143050076615513, −10.27347489171539869436096300256, −8.694764825102331407078890332750, −8.317472231631706629208547189881, −6.46601270199302247999956570867, −5.34906662824903584099103023173, −4.55852930372562267820744144143, −3.18837773879950627308027843800,
1.65526340373510059488696332556, 3.17761174128057410408975713498, 4.85229787696575939833634505152, 6.22904403886653725174728794952, 7.03145909780236109777064955987, 8.422196212728062034899535513109, 9.985639259069870342171832769497, 11.14363410300292825946973154601, 11.56688416189703004988939591343, 12.75849336578892817475443582377