| L(s) = 1 | + 9.55i·5-s − 9.27i·11-s + 4.24·13-s + 15.5i·17-s + 34.8·19-s + 17.7i·23-s − 66.3·25-s + 26.2i·29-s + 32.0·31-s + 55.3·37-s − 38.4i·41-s + 29.3·43-s + 68.2i·47-s + 1.08i·53-s + 88.6·55-s + ⋯ |
| L(s) = 1 | + 1.91i·5-s − 0.843i·11-s + 0.326·13-s + 0.915i·17-s + 1.83·19-s + 0.772i·23-s − 2.65·25-s + 0.904i·29-s + 1.03·31-s + 1.49·37-s − 0.937i·41-s + 0.682·43-s + 1.45i·47-s + 0.0205i·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.910814700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910814700\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 9.55iT - 25T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 4.24T + 169T^{2} \) |
| 17 | \( 1 - 15.5iT - 289T^{2} \) |
| 19 | \( 1 - 34.8T + 361T^{2} \) |
| 23 | \( 1 - 17.7iT - 529T^{2} \) |
| 29 | \( 1 - 26.2iT - 841T^{2} \) |
| 31 | \( 1 - 32.0T + 961T^{2} \) |
| 37 | \( 1 - 55.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 29.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.08iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8.65T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 148.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 88.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 76.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 12.7T + 9.40e3T^{2} \) |
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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
−9.576201072254436427963252598530, −8.527759234309867066272486471951, −7.54675790040656112661314102796, −7.16750694840821125077993524909, −6.00547967871380880639923961157, −5.80648651075254365263452510990, −4.18919288883774549342337071300, −3.21757506687676534129070232299, −2.81213703765693816030004712712, −1.30655400062952247127484198251,
0.56174532089995450249990047182, 1.37791767062607346548234643920, 2.67996748793577988485862282488, 4.09489927392412180157615333567, 4.76733081547474464269974726292, 5.35478497045439804726839410458, 6.31975384679418301992242333378, 7.53044108164906006157258272142, 8.042681467682698428367384345588, 8.925866437590504733372524295439
