L(s) = 1 | + (−2.99 − 0.213i)3-s + (2.77 + 4.81i)5-s + (0.506 + 0.877i)7-s + (8.90 + 1.27i)9-s + (−3.59 − 6.21i)11-s − 21.0i·13-s + (−7.28 − 14.9i)15-s + (5.88 − 10.1i)17-s + (4.15 + 18.5i)19-s + (−1.32 − 2.73i)21-s − 38.5·23-s + (−2.93 + 5.09i)25-s + (−26.3 − 5.71i)27-s + (34.4 + 19.8i)29-s + (47.7 + 27.5i)31-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0710i)3-s + (0.555 + 0.962i)5-s + (0.0723 + 0.125i)7-s + (0.989 + 0.141i)9-s + (−0.326 − 0.565i)11-s − 1.61i·13-s + (−0.485 − 0.999i)15-s + (0.346 − 0.599i)17-s + (0.218 + 0.975i)19-s + (−0.0632 − 0.130i)21-s − 1.67·23-s + (−0.117 + 0.203i)25-s + (−0.977 − 0.211i)27-s + (1.18 + 0.686i)29-s + (1.53 + 0.888i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.388177001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388177001\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.99 + 0.213i)T \) |
| 19 | \( 1 + (-4.15 - 18.5i)T \) |
good | 5 | \( 1 + (-2.77 - 4.81i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.506 - 0.877i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.59 + 6.21i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 21.0iT - 169T^{2} \) |
| 17 | \( 1 + (-5.88 + 10.1i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + 38.5T + 529T^{2} \) |
| 29 | \( 1 + (-34.4 - 19.8i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-47.7 - 27.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 39.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-1.00 + 0.580i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 - 32.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-13.8 + 24.0i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-32.1 + 18.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-3.95 + 2.28i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (15.6 - 27.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 41.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-27.3 - 15.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-56.0 + 97.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 1.05iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-76.0 - 131. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-27.7 + 16.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 136. iT - 9.40e3T^{2} \) |
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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
−10.36350169737420970752962332771, −9.895202388343661662375885880722, −8.326833864407381043065800099341, −7.55322651630577442000261145816, −6.47295998723801684951965484845, −5.82258530513836838087046641732, −5.10335598997722537461950167379, −3.57401606586203894904248043612, −2.44348789518427692799778633289, −0.75365258000663127394507774262,
0.976047381954980540006243946507, 2.14834731932752902417908131871, 4.33467730762845243437603351752, 4.64116576375898782449169603126, 5.86705538356316118049562513144, 6.50850018580135657093769084702, 7.60913632366085480047049049409, 8.691299967323452780939034356251, 9.705798611478660419226545839350, 10.08882611190004072967091275522