L(s) = 1 | + 4i·2-s − 16·4-s + (−55 + 10i)5-s + 158i·7-s − 64i·8-s + (−40 − 220i)10-s + 148·11-s − 684i·13-s − 632·14-s + 256·16-s − 2.04e3i·17-s − 2.22e3·19-s + (880 − 160i)20-s + 592i·22-s − 1.24e3i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.983 + 0.178i)5-s + 1.21i·7-s − 0.353i·8-s + (−0.126 − 0.695i)10-s + 0.368·11-s − 1.12i·13-s − 0.861·14-s + 0.250·16-s − 1.71i·17-s − 1.41·19-s + (0.491 − 0.0894i)20-s + 0.260i·22-s − 0.491i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.314475 - 0.262453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.314475 - 0.262453i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (55 - 10i)T \) |
good | 7 | \( 1 - 158iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 148T + 1.61e5T^{2} \) |
| 13 | \( 1 + 684iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.04e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 270T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.37e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.72e4iT - 8.58e9T^{2} \) |
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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
−12.79815699147473866154743754861, −12.01489114720489222406930790558, −10.81925603484668640903743472011, −9.233753046113321295957484866732, −8.333180019243745791419694862899, −7.23065331322161692210391381178, −5.90046019645149980008757581893, −4.58969947940424697057555459734, −2.89230975449591630649322238711, −0.17294924110467147987588386132,
1.51003421998450268117401362950, 3.76574800022741387248522764430, 4.38166434214523144874307368998, 6.57191011569547316844062799138, 7.889540861626861394864905455711, 8.982643058611419284405275498388, 10.44169349266706109599366942044, 11.15492254074759359287940705201, 12.25527056946384563452647572226, 13.18746933666727150660279706217