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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4627041061\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4627041061\) |
\(L(\frac{7}{2})\) |
|
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
−10.92353089842328945980813996438, −10.47380921232647112719455977234, −9.831574821893549120509441762964, −8.020208461237298252821547620210, −7.47566208883051525136103419685, −6.18287778149971004664818180643, −5.77082532511167198374389294039, −4.45265056064079725816962736900, −2.91628473396909536211726276211, −1.96031781324917911358079000127,
0.21923392094176048168452255103, 0.60733998183980276787130108910, 2.36216601446818196075783058137, 4.28360314478464177249817541791, 5.02745127640623939892665676718, 5.54784115902475033311110302692, 7.18887943648976934730966928019, 7.75231784586297757052349922647, 9.128177811597549425981483178912, 9.942805149550007578572315393124