| L(s) = 1 | − i·2-s − 4-s + 2.09·5-s + (−2.34 + 1.22i)7-s + i·8-s − 2.09i·10-s + 5.08i·11-s − 5.72i·13-s + (1.22 + 2.34i)14-s + 16-s − 0.0838·17-s − 2.68i·19-s − 2.09·20-s + 5.08·22-s − 5.04i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.936·5-s + (−0.887 + 0.461i)7-s + 0.353i·8-s − 0.662i·10-s + 1.53i·11-s − 1.58i·13-s + (0.326 + 0.627i)14-s + 0.250·16-s − 0.0203·17-s − 0.615i·19-s − 0.468·20-s + 1.08·22-s − 1.05i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582030932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582030932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.34 - 1.22i)T \) |
| 29 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.09T + 5T^{2} \) |
| 11 | \( 1 - 5.08iT - 11T^{2} \) |
| 13 | \( 1 + 5.72iT - 13T^{2} \) |
| 17 | \( 1 + 0.0838T + 17T^{2} \) |
| 19 | \( 1 + 2.68iT - 19T^{2} \) |
| 23 | \( 1 + 5.04iT - 23T^{2} \) |
| 31 | \( 1 - 2.19iT - 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 3.12iT - 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 0.169iT - 61T^{2} \) |
| 67 | \( 1 + 0.665T + 67T^{2} \) |
| 71 | \( 1 + 14.8iT - 71T^{2} \) |
| 73 | \( 1 - 3.15iT - 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 7.54T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 7.01iT - 97T^{2} \) |
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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
−8.597252937520346703022045761570, −7.60918971002053921191228775336, −6.79827773348243384616417863094, −5.97418838243084206410250468009, −5.29559634614029762945781319514, −4.53597281744048348377841333655, −3.41741341639616156678979217992, −2.56824635808618300147545011534, −1.99640233645110805885832286149, −0.53443716851376306943713634056,
1.01874536810933312350347668463, 2.25277508804243804542468131593, 3.50550671885584463188179537206, 4.02623460054421760207370983569, 5.25888349038955443976858694962, 5.96480964100389547741825900017, 6.36520216421617688098606904152, 7.12775513285371401108538631051, 7.909487724834322636567681653017, 9.001217870719705071426901879602
