L(s) = 1 | + (−0.460 + 0.634i)2-s + (0.951 + 0.309i)3-s + (0.428 + 1.31i)4-s + (2.23 − 0.0529i)5-s + (−0.634 + 0.460i)6-s + (−1.44 + 0.469i)7-s + (−2.52 − 0.820i)8-s + (0.809 + 0.587i)9-s + (−0.996 + 1.44i)10-s + (0.438 − 3.28i)11-s + 1.38i·12-s + (−0.274 + 0.377i)13-s + (0.367 − 1.13i)14-s + (2.14 + 0.640i)15-s + (−0.557 + 0.404i)16-s + (0.997 + 1.37i)17-s + ⋯ |
L(s) = 1 | + (−0.325 + 0.448i)2-s + (0.549 + 0.178i)3-s + (0.214 + 0.658i)4-s + (0.999 − 0.0236i)5-s + (−0.258 + 0.188i)6-s + (−0.545 + 0.177i)7-s + (−0.892 − 0.290i)8-s + (0.269 + 0.195i)9-s + (−0.315 + 0.456i)10-s + (0.132 − 0.991i)11-s + 0.399i·12-s + (−0.0760 + 0.104i)13-s + (0.0983 − 0.302i)14-s + (0.553 + 0.165i)15-s + (−0.139 + 0.101i)16-s + (0.241 + 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06101 + 0.703796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06101 + 0.703796i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.23 + 0.0529i)T \) |
| 11 | \( 1 + (-0.438 + 3.28i)T \) |
good | 2 | \( 1 + (0.460 - 0.634i)T + (-0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (1.44 - 0.469i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.274 - 0.377i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.997 - 1.37i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.73 - 5.33i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 7.68iT - 23T^{2} \) |
| 29 | \( 1 + (0.822 + 2.53i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.20 + 2.32i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.90 + 0.945i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.417 - 1.28i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + (-2.43 - 0.790i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (4.71 - 6.48i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.33 + 7.17i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.99 + 6.53i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 6.89iT - 67T^{2} \) |
| 71 | \( 1 + (11.8 - 8.57i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.15 - 0.700i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.85 - 7.15i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.92 - 9.53i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 1.47T + 89T^{2} \) |
| 97 | \( 1 + (8.34 - 11.4i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.99509472286470345369396612421, −12.30968368242640403727158934632, −10.81099853154687021935751391078, −9.738421717086263311366202294367, −8.824901996406899760090162751739, −8.067204108119645702625966354156, −6.63849663980596490254642506194, −5.82631538329772523294190820066, −3.79335157854029511334633783471, −2.49769725787335341054600188033,
1.66640248454065349936614397650, 2.95122039322562884942638140900, 5.03497728175245110321192514798, 6.33098019926684836371674082469, 7.29869244417575322988715789591, 9.095898055038949846804474177857, 9.572003260476750002010087859558, 10.38137888423307244922202452508, 11.52150379550333391288134284633, 12.79620834336701703998569834923