| L(s) = 1 | − 1.73i·3-s − 1.36·5-s + 1.24i·7-s − 2.99·9-s − 5.79i·11-s − 16.3·13-s + 2.36i·15-s − 5.01·17-s − 26.1i·19-s + 2.15·21-s + 25.1i·23-s − 23.1·25-s + 5.19i·27-s − 32.7·29-s − 1.01i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.272·5-s + 0.177i·7-s − 0.333·9-s − 0.527i·11-s − 1.26·13-s + 0.157i·15-s − 0.294·17-s − 1.37i·19-s + 0.102·21-s + 1.09i·23-s − 0.925·25-s + 0.192i·27-s − 1.13·29-s − 0.0328i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.387459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.387459i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 + 1.36T + 25T^{2} \) |
| 7 | \( 1 - 1.24iT - 49T^{2} \) |
| 11 | \( 1 + 5.79iT - 121T^{2} \) |
| 13 | \( 1 + 16.3T + 169T^{2} \) |
| 17 | \( 1 + 5.01T + 289T^{2} \) |
| 19 | \( 1 + 26.1iT - 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 + 32.7T + 841T^{2} \) |
| 31 | \( 1 + 1.01iT - 961T^{2} \) |
| 37 | \( 1 + 14.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 111.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−10.85930232711769825897690737037, −9.614100780859093532015308584761, −8.832880603528543203403240703843, −7.67600751848302264065023411908, −7.06433642904175609275250363690, −5.84069539942194259371365827844, −4.81810258357339560174812460948, −3.33467711206721674766703980120, −2.04243914786588183805626238513, −0.15882316132642915296889295001,
2.13923115926971451085715332291, 3.66026410980715678240478978575, 4.61382630628657586024668580620, 5.66237900625439461758041479248, 6.97072975888666649928548577652, 7.86173516264167821841222143465, 8.892518691346738372718414373094, 9.991645768504340705699203906164, 10.37741792594436060267428245855, 11.69224411299775875235950635362
