| L(s) = 1 | + 482i·3-s − 1.73e5·9-s − 9.74e4i·11-s + 8.23e5·17-s − 3.35e6i·19-s − 9.76e6·25-s − 5.50e7i·27-s + 4.69e7·33-s + 3.77e7·41-s + 2.14e8i·43-s + 2.82e8·49-s + 3.97e8i·51-s + 1.61e9·57-s + 9.21e8i·59-s − 1.81e9i·67-s + ⋯ |
| L(s) = 1 | + 1.98i·3-s − 2.93·9-s − 0.604i·11-s + 0.580·17-s − 1.35i·19-s − 25-s − 3.83i·27-s + 1.19·33-s + 0.326·41-s + 1.45i·43-s + 49-s + 1.15i·51-s + 2.68·57-s + 1.28i·59-s − 1.34i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.689239016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689239016\) |
| \(L(6)\) |
|
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 | \( 1 - 482iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 9.76e6T^{2} \) |
| 7 | \( 1 - 2.82e8T^{2} \) |
| 11 | \( 1 + 9.74e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 1.37e11T^{2} \) |
| 17 | \( 1 - 8.23e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 3.35e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 4.14e13T^{2} \) |
| 29 | \( 1 + 4.20e14T^{2} \) |
| 31 | \( 1 - 8.19e14T^{2} \) |
| 37 | \( 1 + 4.80e15T^{2} \) |
| 41 | \( 1 - 3.77e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 2.14e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 5.25e16T^{2} \) |
| 53 | \( 1 + 1.74e17T^{2} \) |
| 59 | \( 1 - 9.21e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.81e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.60e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 + 9.60e7iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 1.11e10T + 3.11e19T^{2} \) |
| 97 | \( 1 + 9.87e9T + 7.37e19T^{2} \) |
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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
−10.44996959375959412656498383240, −9.530246646998225052355593058030, −8.928324437181341910535031516361, −7.889482939787529280966031084479, −6.17876630079453317565723370794, −5.29156774411664826667751814231, −4.41341621863886440883815292058, −3.48541049697589242675472413985, −2.60438409491548692666674371807, −0.55442742998783047365003468062,
0.57064901195125456626994626102, 1.59807298254026733413003745800, 2.30727010003834163071107601618, 3.61166245656403964721368432477, 5.43549409507043161827830056186, 6.19127811717823210904449633741, 7.22149046588938741939410715390, 7.82142824514339963800788591670, 8.689795877256466681096002823846, 9.999267421130649003589759234443
