L(s) = 1 | − 2.16i·2-s + (−1.71 + 0.256i)3-s − 2.67·4-s + 0.517·5-s + (0.553 + 3.70i)6-s + (1.45 − 2.20i)7-s + 1.44i·8-s + (2.86 − 0.878i)9-s − 1.11i·10-s − 4.77i·11-s + (4.57 − 0.684i)12-s + 1.14i·13-s + (−4.76 − 3.15i)14-s + (−0.886 + 0.132i)15-s − 2.20·16-s − 1.79·17-s + ⋯ |
L(s) = 1 | − 1.52i·2-s + (−0.988 + 0.147i)3-s − 1.33·4-s + 0.231·5-s + (0.226 + 1.51i)6-s + (0.551 − 0.833i)7-s + 0.512i·8-s + (0.956 − 0.292i)9-s − 0.353i·10-s − 1.44i·11-s + (1.32 − 0.197i)12-s + 0.317i·13-s + (−1.27 − 0.843i)14-s + (−0.228 + 0.0342i)15-s − 0.552·16-s − 0.435·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.184663 + 0.833687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.184663 + 0.833687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 - 0.256i)T \) |
| 7 | \( 1 + (-1.45 + 2.20i)T \) |
| 23 | \( 1 + iT \) |
good | 2 | \( 1 + 2.16iT - 2T^{2} \) |
| 5 | \( 1 - 0.517T + 5T^{2} \) |
| 11 | \( 1 + 4.77iT - 11T^{2} \) |
| 13 | \( 1 - 1.14iT - 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 - 3.65iT - 19T^{2} \) |
| 29 | \( 1 - 2.33iT - 29T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 37 | \( 1 + 2.59T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 - 0.121iT - 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 9.95iT - 61T^{2} \) |
| 67 | \( 1 - 4.00T + 67T^{2} \) |
| 71 | \( 1 - 8.01iT - 71T^{2} \) |
| 73 | \( 1 + 3.71iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 9.95T + 89T^{2} \) |
| 97 | \( 1 - 3.44iT - 97T^{2} \) |
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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
−10.67106182335704472554202289174, −10.06872721414518951776380042209, −9.111953632206117186807350353892, −7.88092312230054206453171639754, −6.57419053821823494634944211957, −5.54928797064391188480625484222, −4.31645094267353798000812684368, −3.59989069130470957186472802437, −1.86910519555662937586941423927, −0.59304964506342973643817174255,
2.03637368776800789862358472608, 4.52895126264754952175323988265, 5.14692695918564796628750464804, 5.94854054404298213218593016783, 6.89997845958869635246582826845, 7.49612200135948441302796633929, 8.587420020364758353992919054212, 9.498554537526329248235914722767, 10.54290235567565735572523023965, 11.62465907893858668763074932193