L(s) = 1 | − 1.41·2-s + 2.00·4-s − 4i·7-s − 2.82·8-s + 16.9i·11-s − 8i·13-s + 5.65i·14-s + 4.00·16-s + 12.7·17-s + 16·19-s − 24i·22-s − 16.9·23-s + 11.3i·26-s − 8.00i·28-s − 4.24i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.571i·7-s − 0.353·8-s + 1.54i·11-s − 0.615i·13-s + 0.404i·14-s + 0.250·16-s + 0.748·17-s + 0.842·19-s − 1.09i·22-s − 0.737·23-s + 0.435i·26-s − 0.285i·28-s − 0.146i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.247696675\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247696675\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4iT - 49T^{2} \) |
| 11 | \( 1 - 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 - 12.7T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + 16.9T + 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176iT - 9.40e3T^{2} \) |
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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
−10.57451177867363531473430131930, −10.00442067561812546474786042145, −9.314309969026478604215550419163, −7.935418760502334843341108974140, −7.50731261588807554406179894904, −6.44522557974058556748731550778, −5.19847989824462011531725691800, −3.96152588593693617168521800480, −2.50287939434034797799847436192, −1.01128128156617840631028310776,
0.897966357199475451636964287621, 2.53426539421264637857551990932, 3.71734153533127080493838884752, 5.41490141138057600432018156025, 6.16169065051957684704526866819, 7.31578790291450245221999301900, 8.356082434131795040344992808122, 8.917847320754292213343374260950, 9.925544789066230034188432678328, 10.75414019133472574003758606596