| L(s) = 1 | − 1.73i·3-s + 5.83·5-s − 5.24·7-s − 2.99·9-s − 1.24·11-s − 5.89i·13-s − 10.1i·15-s − 1.57·17-s + (−10.6 − 15.7i)19-s + 9.07i·21-s − 27.5·23-s + 9.05·25-s + 5.19i·27-s + 15.9i·29-s − 53.2i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.16·5-s − 0.748·7-s − 0.333·9-s − 0.112·11-s − 0.453i·13-s − 0.673i·15-s − 0.0923·17-s + (−0.561 − 0.827i)19-s + 0.432i·21-s − 1.19·23-s + 0.362·25-s + 0.192i·27-s + 0.548i·29-s − 1.71i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.176250468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176250468\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (10.6 + 15.7i)T \) |
| good | 5 | \( 1 - 5.83T + 25T^{2} \) |
| 7 | \( 1 + 5.24T + 49T^{2} \) |
| 11 | \( 1 + 1.24T + 121T^{2} \) |
| 13 | \( 1 + 5.89iT - 169T^{2} \) |
| 17 | \( 1 + 1.57T + 289T^{2} \) |
| 23 | \( 1 + 27.5T + 529T^{2} \) |
| 29 | \( 1 - 15.9iT - 841T^{2} \) |
| 31 | \( 1 + 53.2iT - 961T^{2} \) |
| 37 | \( 1 + 10.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 40.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 75.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 47.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 56.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 24.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.19iT - 9.40e3T^{2} \) |
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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
−9.563022310872726375417267200494, −8.850559842421169893093922164942, −7.79918567545154716067512868637, −6.87347250447306958768218425875, −6.04459158094007122208958813109, −5.52744426969276324913226095412, −4.09737785632460629895641032874, −2.74049306947625592249865551901, −1.93591309105711078343027722240, −0.34778027883780155609669672267,
1.67889518065885754144592047254, 2.82330410639790283968937586084, 3.94866666676744572030419100220, 4.99657632661111438120043478562, 6.10205913801720196657277502785, 6.40418928561681697530967797483, 7.80260569372349849604581239180, 8.797864822351301656103810376969, 9.604032323113662973170978903913, 10.06164781455664220778452663135
