| L(s) = 1 | + (−1.99 + 0.194i)2-s + (−0.136 + 2.99i)3-s + (3.92 − 0.773i)4-s + (−3.84 + 2.22i)5-s + (−0.311 − 5.99i)6-s + (−0.704 − 0.406i)7-s + (−7.66 + 2.30i)8-s + (−8.96 − 0.816i)9-s + (7.22 − 5.16i)10-s + (−3.72 + 6.44i)11-s + (1.78 + 11.8i)12-s + (−18.0 + 10.4i)13-s + (1.48 + 0.672i)14-s + (−6.13 − 11.8i)15-s + (14.8 − 6.07i)16-s + 1.74·17-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0971i)2-s + (−0.0453 + 0.998i)3-s + (0.981 − 0.193i)4-s + (−0.769 + 0.444i)5-s + (−0.0518 − 0.998i)6-s + (−0.100 − 0.0580i)7-s + (−0.957 + 0.287i)8-s + (−0.995 − 0.0906i)9-s + (0.722 − 0.516i)10-s + (−0.338 + 0.585i)11-s + (0.148 + 0.988i)12-s + (−1.39 + 0.803i)13-s + (0.105 + 0.0480i)14-s + (−0.408 − 0.788i)15-s + (0.925 − 0.379i)16-s + 0.102·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.119678 + 0.471559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119678 + 0.471559i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 - 0.194i)T \) |
| 3 | \( 1 + (0.136 - 2.99i)T \) |
| good | 5 | \( 1 + (3.84 - 2.22i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.704 + 0.406i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.72 - 6.44i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (18.0 - 10.4i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.74T + 289T^{2} \) |
| 19 | \( 1 - 31.7T + 361T^{2} \) |
| 23 | \( 1 + (-6.44 + 3.72i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-26.9 - 15.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-4.91 + 2.83i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 62.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-2.74 - 4.74i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.3 - 38.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-71.1 - 41.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 85.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (21.8 + 37.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (61.1 + 35.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-9.91 - 17.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 69.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-37.1 - 21.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (3.53 - 6.11i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 37.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-9.38 + 16.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.16194777050762397531949107616, −14.30116478839072256823474248901, −12.05349995860122277892194093185, −11.37412624357915381230278553287, −10.09946320512756040240641518345, −9.456704030772009833968203492613, −7.969276004608174083050267141003, −6.89382166563505238064893983295, −4.95832644321228538419306008102, −3.02549699573486377201200510054,
0.56741437159204504098802948696, 2.84808349423558564962575313593, 5.62652027464830593684983840280, 7.34863851594519664713187751213, 7.896032461540901353904899062370, 9.136853870582094699890881432358, 10.59515208824342947800173713060, 11.96589490470417404980065084997, 12.31067610207416643854114615884, 13.82444935439258099619556306835
