| L(s) = 1 | − 5.79i·5-s − 6.19·7-s − 1.13i·11-s − 11.3·13-s + 31.2i·17-s − 32.9·19-s − 33.6i·23-s − 8.58·25-s + 25.5i·29-s − 23.1·31-s + 35.9i·35-s + 8.80·37-s − 70.6i·41-s − 59.7·43-s − 60.2i·47-s + ⋯ |
| L(s) = 1 | − 1.15i·5-s − 0.885·7-s − 0.103i·11-s − 0.876·13-s + 1.83i·17-s − 1.73·19-s − 1.46i·23-s − 0.343·25-s + 0.881i·29-s − 0.747·31-s + 1.02i·35-s + 0.237·37-s − 1.72i·41-s − 1.38·43-s − 1.28i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.330735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.330735i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.79iT - 25T^{2} \) |
| 7 | \( 1 + 6.19T + 49T^{2} \) |
| 11 | \( 1 + 1.13iT - 121T^{2} \) |
| 13 | \( 1 + 11.3T + 169T^{2} \) |
| 17 | \( 1 - 31.2iT - 289T^{2} \) |
| 19 | \( 1 + 32.9T + 361T^{2} \) |
| 23 | \( 1 + 33.6iT - 529T^{2} \) |
| 29 | \( 1 - 25.5iT - 841T^{2} \) |
| 31 | \( 1 + 23.1T + 961T^{2} \) |
| 37 | \( 1 - 8.80T + 1.36e3T^{2} \) |
| 41 | \( 1 + 70.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 60.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 30.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 10.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 3.63iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 11.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74.3T + 9.40e3T^{2} \) |
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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
−10.71228999433244646365040697200, −10.08266428274853886885797748564, −8.774977640478964621911303032731, −8.482452836500641620137724374713, −6.94117564061513007930714064398, −6.00118198336076197412630527494, −4.80146518106209240470120357303, −3.76764248081945832827513810956, −2.03011187513671819363068225277, −0.14391730030010918913708213884,
2.43281794703789839243783158849, 3.36825784832275787432076209081, 4.84260307966605050556516219645, 6.26785315456034646215761547319, 6.95783733083999783905054535917, 7.84660189974368037869576414727, 9.424133072773007097578932067763, 9.857436918893879998122074371425, 10.97938203165922393796520604197, 11.69728740129530970417013330454
