L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.35 − 1.08i)3-s + (0.499 + 0.866i)4-s + 5-s + (0.630 + 1.61i)6-s + (−1.80 + 1.93i)7-s − 0.999i·8-s + (0.657 + 2.92i)9-s + (−0.866 − 0.5i)10-s − 0.781i·11-s + (0.260 − 1.71i)12-s + (−2.26 − 1.31i)13-s + (2.52 − 0.774i)14-s + (−1.35 − 1.08i)15-s + (−0.5 + 0.866i)16-s + (3.02 − 5.24i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.780 − 0.624i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.257 + 0.658i)6-s + (−0.681 + 0.731i)7-s − 0.353i·8-s + (0.219 + 0.975i)9-s + (−0.273 − 0.158i)10-s − 0.235i·11-s + (0.0753 − 0.494i)12-s + (−0.629 − 0.363i)13-s + (0.676 − 0.206i)14-s + (−0.349 − 0.279i)15-s + (−0.125 + 0.216i)16-s + (0.734 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.443905 - 0.558599i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.443905 - 0.558599i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.35 + 1.08i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.80 - 1.93i)T \) |
good | 11 | \( 1 + 0.781iT - 11T^{2} \) |
| 13 | \( 1 + (2.26 + 1.31i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 5.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.09 + 3.51i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4.60iT - 23T^{2} \) |
| 29 | \( 1 + (-3.96 + 2.28i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.47 - 3.74i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.42 + 9.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.46 + 9.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.45 + 7.70i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.501 + 0.868i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.30 - 4.79i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.32 - 2.28i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.84 - 1.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.02 + 10.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (-7.67 - 4.42i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.96 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.03 + 3.52i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.78 + 6.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.00 + 2.88i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.30890281046171739913090542457, −9.527031815065787818914698039773, −8.843734877977047637370777149508, −7.37141483379211134278881214556, −7.12185525859168799755286293444, −5.65110493892401722436115137559, −5.27647506720904413868914682674, −3.23987619915442324315656494814, −2.19919959138182678682870104982, −0.59367403926084281956554422918,
1.25517302375833799860909737046, 3.27008639943174213731642903599, 4.48558475133516979140909768278, 5.57396389047789424783290525329, 6.38344214942938316532503806033, 7.15175614597198676048100916099, 8.246180387867273079430401626017, 9.481772215583059693150424274580, 10.00783366196553014525936359760, 10.42032670056685733788621279448