L(s) = 1 | − i·2-s + (0.866 − 1.5i)3-s − 4-s + (0.866 − 0.5i)5-s + (−1.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + i·8-s + (−1.5 − 2.59i)9-s + (−0.5 − 0.866i)10-s + (0.866 + 1.5i)11-s + (−0.866 + 1.5i)12-s + (6 − 3.46i)13-s + (−0.866 − 0.5i)14-s − 1.73i·15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.499 − 0.866i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.612 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.158 − 0.273i)10-s + (0.261 + 0.452i)11-s + (−0.249 + 0.433i)12-s + (1.66 − 0.960i)13-s + (−0.231 − 0.133i)14-s − 0.447i·15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.620551 - 1.85370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.620551 - 1.85370i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (5.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-6 + 3.46i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 37 | \( 1 + (3 + 1.73i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 1.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + (-0.866 - 1.5i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.59 + 1.5i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9 - 5.19i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.06 - 10.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.575720411928472473615402152217, −8.988478196125720127643823071906, −8.143226258333325712173679094222, −7.37672930685797653753940606887, −6.25265413389042287008658038703, −5.43406098291162095928695456054, −4.02188161281827278587317527763, −3.16183150049974410422541626558, −1.90433603796962316475727073809, −0.949771475103550467207921256036,
1.83271575125807837432533801377, 3.42743978609614313930242761098, 4.06881064201459951498521000381, 5.33624226306567039393140566528, 5.96047659512279719955890125702, 6.96205072517934899884082818252, 8.082456365044019681398376858150, 8.883851088343908879331625429252, 9.204724708558072204728073716928, 10.28713619856137285217404575419