L(s) = 1 | + (2.99 + 0.114i)3-s + (0.801 − 1.38i)7-s + (8.97 + 0.687i)9-s + (−6.10 − 3.52i)11-s + (10.9 + 18.9i)13-s + 33.1i·17-s − 6.82·19-s + (2.56 − 4.06i)21-s + (−12.2 + 7.09i)23-s + (26.8 + 3.08i)27-s + (31.4 + 18.1i)29-s + (−17.2 − 29.8i)31-s + (−17.8 − 11.2i)33-s + 43.1·37-s + (30.5 + 57.9i)39-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0382i)3-s + (0.114 − 0.198i)7-s + (0.997 + 0.0763i)9-s + (−0.554 − 0.320i)11-s + (0.840 + 1.45i)13-s + 1.94i·17-s − 0.359·19-s + (0.121 − 0.193i)21-s + (−0.534 + 0.308i)23-s + (0.993 + 0.114i)27-s + (1.08 + 0.626i)29-s + (−0.555 − 0.961i)31-s + (−0.542 − 0.341i)33-s + 1.16·37-s + (0.784 + 1.48i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.764275696\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.764275696\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.99 - 0.114i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.801 + 1.38i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (6.10 + 3.52i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-10.9 - 18.9i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 33.1iT - 289T^{2} \) |
| 19 | \( 1 + 6.82T + 361T^{2} \) |
| 23 | \( 1 + (12.2 - 7.09i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-31.4 - 18.1i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (17.2 + 29.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 43.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-18.2 + 10.5i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-30.6 + 53.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (14.6 + 8.44i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 12.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-54.3 + 31.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-3.38 + 5.86i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-53.4 - 92.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 23.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (27.8 - 48.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (71.6 + 41.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 13.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-64.6 + 112. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.00970944053297906160162845818, −8.952080020228091849860120552624, −8.467316010976011745268498529319, −7.66761953763675681494441649492, −6.65735007784848881168200427678, −5.79205111498156098948483039266, −4.22208301575139577895813166519, −3.85503075497074905613905606157, −2.42832646447069242546815508454, −1.43324460221454389831475047811,
0.855771147318606570389527735011, 2.46913862894426241376245598098, 3.10693115538157265032206174752, 4.37766731922717025464585252824, 5.31148701797852632575363688088, 6.47283997285380358878282016594, 7.54939880329794819824948395722, 8.083164895153943477535024655611, 8.898572867281523390503055800343, 9.779285616158361694590655060218