L(s) = 1 | + (−1.85 + 0.756i)2-s + (2.85 − 2.80i)4-s − 1.08·5-s − 6.01i·7-s + (−3.16 + 7.34i)8-s + (2.00 − 0.817i)10-s − 17.7i·11-s + 12.4·13-s + (4.55 + 11.1i)14-s + (0.291 − 15.9i)16-s + 26.3·17-s − 5.19i·19-s + (−3.08 + 3.02i)20-s + (13.4 + 32.8i)22-s − 29.8i·23-s + ⋯ |
L(s) = 1 | + (−0.925 + 0.378i)2-s + (0.713 − 0.700i)4-s − 0.216·5-s − 0.859i·7-s + (−0.395 + 0.918i)8-s + (0.200 − 0.0817i)10-s − 1.61i·11-s + 0.955·13-s + (0.325 + 0.795i)14-s + (0.0182 − 0.999i)16-s + 1.55·17-s − 0.273i·19-s + (−0.154 + 0.151i)20-s + (0.609 + 1.49i)22-s − 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.778873 - 0.326788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.778873 - 0.326788i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.85 - 0.756i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.08T + 25T^{2} \) |
| 7 | \( 1 + 6.01iT - 49T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 - 12.4T + 169T^{2} \) |
| 17 | \( 1 - 26.3T + 289T^{2} \) |
| 19 | \( 1 + 5.19iT - 361T^{2} \) |
| 23 | \( 1 + 29.8iT - 529T^{2} \) |
| 29 | \( 1 + 4.32T + 841T^{2} \) |
| 31 | \( 1 - 44.8iT - 961T^{2} \) |
| 37 | \( 1 + 20.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 59.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 28.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 18.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 94.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 35.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 76.3T + 9.40e3T^{2} \) |
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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
−13.66389103193717761275633435310, −11.99183742331586692069430029149, −10.89447437147623304328939601693, −10.24583734347867563678763314653, −8.759568701739627499578848772623, −8.013378396770474817179446458214, −6.74054889543405863714129766889, −5.57929281644472085540029494791, −3.44725267755043768281430803480, −0.906628958824251358634793561754,
1.85612662493467579453232115499, 3.65476343002419862204011902697, 5.71535192076209459497135949131, 7.26853413320987642076573198191, 8.202070679685298636173494516849, 9.460142735407644260191542772812, 10.15891517653557210210134346580, 11.65999936052383601389167160090, 12.13024834166871186075512697189, 13.30683190271931658251902887884