L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−1.5 + 2.59i)7-s − 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (1.5 − 2.59i)14-s + 0.999·15-s + 16-s + (−1 + 1.73i)17-s + (0.499 + 0.866i)18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.566 + 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s + (0.400 − 0.694i)14-s + 0.258·15-s + 0.250·16-s + (−0.242 + 0.420i)17-s + (0.117 + 0.204i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.459294 + 0.609122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.459294 + 0.609122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-3.5 - 4.33i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.990708414421564382708284190724, −9.554737234997355804079623133824, −8.557957725351166902086562166440, −8.015751975153089056442500790047, −6.83945226507378624630101112173, −6.33257837740565716810340677601, −5.38111751583812566778080204148, −3.70186392474151333235598383124, −2.61755259340189275410882636014, −1.70649617039221832800780428782,
0.41405623119766706184646915098, 2.07000083139532324582035261716, 3.42660656730721717620560887502, 4.29530707641256811094705599474, 5.52310704239098164005590905895, 6.56099427724685621792942273860, 7.40800210902523445082908965929, 8.282725662691111679503335049537, 9.142657534957722120597509163953, 9.752915314096384660635689240277