| L(s) = 1 | + (15.4 − 2.16i)3-s + 81.6i·5-s − 91.7i·7-s + (233. − 66.7i)9-s + 710.·11-s + 666.·13-s + (176. + 1.25e3i)15-s − 2.17e3i·17-s + 1.12e3i·19-s + (−198. − 1.41e3i)21-s − 3.12e3·23-s − 3.53e3·25-s + (3.46e3 − 1.53e3i)27-s − 371. i·29-s − 2.50e3i·31-s + ⋯ |
| L(s) = 1 | + (0.990 − 0.138i)3-s + 1.45i·5-s − 0.707i·7-s + (0.961 − 0.274i)9-s + 1.77·11-s + 1.09·13-s + (0.202 + 1.44i)15-s − 1.82i·17-s + 0.715i·19-s + (−0.0981 − 0.700i)21-s − 1.22·23-s − 1.13·25-s + (0.914 − 0.405i)27-s − 0.0819i·29-s − 0.468i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.831361640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.831361640\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.4 + 2.16i)T \) |
| good | 5 | \( 1 - 81.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 91.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 710.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 666.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.17e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.12e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 371. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.50e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 6.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.54e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.36e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.37e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.47e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.27e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.87e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e4T + 8.58e9T^{2} \) |
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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
−10.42815144708941952846189190407, −9.632818384342173044552618895708, −8.759657587155892506926909040109, −7.51954963559599077403052670359, −6.95718226006445614901652089870, −6.12298131875863391970899439747, −4.02628839931834421016333071214, −3.57803922984512494419643752703, −2.35016859724235641813676767870, −1.04819452899975484955622706373,
1.15974720899120501801446705749, 1.89621122359005009797287002025, 3.71109022810229607428076211847, 4.26100206869800558609768537096, 5.64839588306101753943361133596, 6.66798499064899894846392216693, 8.270426515705430160666695879492, 8.673950754848488558330582189193, 9.184011897419329811945743629994, 10.24837272866055470062635182314
