L(s) = 1 | + (0.0669 − 1.41i)2-s + (−0.665 − 1.15i)3-s + (−1.99 − 0.189i)4-s + (−0.5 − 0.866i)5-s + (−1.67 + 0.862i)6-s − 4.68i·7-s + (−0.400 + 2.79i)8-s + (0.615 − 1.06i)9-s + (−1.25 + 0.648i)10-s + 4.21i·11-s + (1.10 + 2.41i)12-s + (−2.65 − 1.53i)13-s + (−6.62 − 0.313i)14-s + (−0.665 + 1.15i)15-s + (3.92 + 0.752i)16-s + (0.913 + 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.0473 − 0.998i)2-s + (−0.384 − 0.665i)3-s + (−0.995 − 0.0945i)4-s + (−0.223 − 0.387i)5-s + (−0.682 + 0.352i)6-s − 1.77i·7-s + (−0.141 + 0.989i)8-s + (0.205 − 0.355i)9-s + (−0.397 + 0.205i)10-s + 1.27i·11-s + (0.319 + 0.698i)12-s + (−0.737 − 0.425i)13-s + (−1.77 − 0.0838i)14-s + (−0.171 + 0.297i)15-s + (0.982 + 0.188i)16-s + (0.221 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.262309 + 0.739837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.262309 + 0.739837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0669 + 1.41i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-2.30 - 3.69i)T \) |
good | 3 | \( 1 + (0.665 + 1.15i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 4.68iT - 7T^{2} \) |
| 11 | \( 1 - 4.21iT - 11T^{2} \) |
| 13 | \( 1 + (2.65 + 1.53i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.913 - 1.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (5.23 + 3.02i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 2.54iT - 37T^{2} \) |
| 41 | \( 1 + (-10.9 + 6.32i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.19 + 4.15i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.93 - 4.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.880 + 0.508i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.293 + 0.508i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.20 + 7.28i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.92 - 3.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.658 + 1.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.62 + 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.76iT - 83T^{2} \) |
| 89 | \( 1 + (5.73 + 3.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.590 - 0.340i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.71581414380509161980188735156, −10.12159375039766891710028956034, −9.325028272873224052099044346981, −7.63919318099692274215282848859, −7.42385888976156352919115153526, −5.83204230553564969253707247466, −4.41827370639328888072808701124, −3.81619668399651383674543122036, −1.83902049580878375275093915539, −0.55583313546375419381871464760,
2.80069098332678333370757891664, 4.26504941032878113883645450611, 5.49215707295782530397184123671, 5.80511015644376848160178205057, 7.20725285656335683826548287095, 8.181348996898425654563475826586, 9.181473348131196759590112588503, 9.725912340529727991414380874324, 11.10369161902910444052855015851, 11.79089264593863200772732670379