L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.69 + 0.338i)3-s + (0.499 − 0.866i)4-s + 5-s + (−1.30 + 1.14i)6-s + (−2.46 − 0.968i)7-s − 0.999i·8-s + (2.77 − 1.14i)9-s + (0.866 − 0.5i)10-s + 0.656i·11-s + (−0.556 + 1.64i)12-s + (2.39 − 1.38i)13-s + (−2.61 + 0.392i)14-s + (−1.69 + 0.338i)15-s + (−0.5 − 0.866i)16-s + (−3.36 − 5.83i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.980 + 0.195i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (−0.531 + 0.466i)6-s + (−0.930 − 0.366i)7-s − 0.353i·8-s + (0.923 − 0.382i)9-s + (0.273 − 0.158i)10-s + 0.197i·11-s + (−0.160 + 0.473i)12-s + (0.664 − 0.383i)13-s + (−0.699 + 0.104i)14-s + (−0.438 + 0.0873i)15-s + (−0.125 − 0.216i)16-s + (−0.817 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.764755 - 1.01891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.764755 - 1.01891i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (1.69 - 0.338i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.46 + 0.968i)T \) |
good | 11 | \( 1 - 0.656iT - 11T^{2} \) |
| 13 | \( 1 + (-2.39 + 1.38i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.36 + 5.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.633 + 0.365i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.20iT - 23T^{2} \) |
| 29 | \( 1 + (-6.21 - 3.58i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.93 + 4.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 1.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.35 + 5.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.29 + 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.92 - 5.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.77 - 3.33i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.62 - 8.01i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 0.782i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 5.68i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.54iT - 71T^{2} \) |
| 73 | \( 1 + (-5.95 + 3.43i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.53 - 13.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.71 - 15.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.32 - 2.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.02 - 2.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.72698425831082716423450098068, −9.700179066512592266035691885880, −8.960398785412115955895941650033, −7.18863478751320417000566732644, −6.57102619018359634914030971982, −5.74100338356442287037740165363, −4.78394030906456846190593415292, −3.83395407126402897040374374468, −2.50455799624188299756295917715, −0.64485933328790679446040362447,
1.75640649574337628064271438361, 3.39619501985551884135412596046, 4.49533134882874698583401424956, 5.68648558578527632230720974820, 6.22729483889617048057015438078, 6.83034741437459377116667310768, 8.080164612493009630455457192027, 9.164723038344995548558915941003, 10.11873544397959561010992637818, 11.02403763804122325007378681225