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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.11062 + 0.885547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11062 + 0.885547i\) |
\(L(2)\) |
|
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
−13.58453040795796605662878550081, −12.83903099616182611584175394735, −11.53777628373762989960932588043, −10.90596942032150664069580008750, −8.948018946934104578126878769535, −8.137571428118512325669404601399, −6.34848310842392049074604568836, −5.77992919134299362254660276308, −4.12254795745738585339832000858, −2.56881973252412563258432380126,
1.84583082074051754470907594465, 3.80552935146742912679130076094, 4.89645823495769108944884924577, 6.47030170286060828515316692291, 7.40617271683920758325301918328, 9.328903328624172675902282436627, 10.42000998144019217589117003495, 11.27295222025833764379652552458, 12.46772156375158257145802730569, 13.40665944388821051950095633950