| L(s) = 1 | − 8.43i·3-s − 12.2i·5-s + 1.63·7-s − 44.1·9-s − 25.7i·11-s − 13.2i·13-s − 103.·15-s − 53.6·17-s − 100. i·19-s − 13.8i·21-s + 25.1·23-s − 25.6·25-s + 144. i·27-s + 256. i·29-s − 132.·31-s + ⋯ |
| L(s) = 1 | − 1.62i·3-s − 1.09i·5-s + 0.0885·7-s − 1.63·9-s − 0.705i·11-s − 0.282i·13-s − 1.78·15-s − 0.764·17-s − 1.21i·19-s − 0.143i·21-s + 0.227·23-s − 0.205·25-s + 1.03i·27-s + 1.63i·29-s − 0.768·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9820418801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9820418801\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8.43iT - 27T^{2} \) |
| 5 | \( 1 + 12.2iT - 125T^{2} \) |
| 7 | \( 1 - 1.63T + 343T^{2} \) |
| 11 | \( 1 + 25.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 13.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 25.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 247. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 404. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 743. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 65.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 273. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 399. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 580. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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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
−8.663228781006056652254386103536, −8.223471889216432682870093279136, −7.17041553953905429607569233391, −6.59541682055860781212958030415, −5.52866143345933738196774448047, −4.79018600126252228072690965714, −3.25462457227019725472584664293, −2.06514178200040955152436996738, −1.08025876769131984641955543924, −0.26046297225256613810190775547,
2.09122357968923892892912870698, 3.20860906231784754342236606399, 4.02725219108821243861276607154, 4.77446400849893411652263992930, 5.84021519814068539550462734304, 6.72992334621177163150659736711, 7.73097347873206393847731724933, 8.771588520614183767213130094315, 9.602809701004325945461090355699, 10.20499236509207338701005102492
