| L(s) = 1 | + (−0.188 + 1.72i)3-s − 2.77i·5-s − 4.28i·7-s + (−2.92 − 0.650i)9-s − 3.54·11-s + 4.92·13-s + (4.77 + 0.523i)15-s + 5.04i·17-s + i·19-s + (7.37 + 0.808i)21-s − 5.26·23-s − 2.69·25-s + (1.67 − 4.91i)27-s − 6.16i·29-s − 1.60i·31-s + ⋯ |
| L(s) = 1 | + (−0.108 + 0.994i)3-s − 1.24i·5-s − 1.61i·7-s + (−0.976 − 0.216i)9-s − 1.06·11-s + 1.36·13-s + (1.23 + 0.135i)15-s + 1.22i·17-s + 0.229i·19-s + (1.60 + 0.176i)21-s − 1.09·23-s − 0.539·25-s + (0.321 − 0.946i)27-s − 1.14i·29-s − 0.288i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7144162833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7144162833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.188 - 1.72i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.77iT - 5T^{2} \) |
| 7 | \( 1 + 4.28iT - 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 17 | \( 1 - 5.04iT - 17T^{2} \) |
| 23 | \( 1 + 5.26T + 23T^{2} \) |
| 29 | \( 1 + 6.16iT - 29T^{2} \) |
| 31 | \( 1 + 1.60iT - 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 - 4.50iT - 41T^{2} \) |
| 43 | \( 1 + 6.51iT - 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 3.80iT - 53T^{2} \) |
| 59 | \( 1 + 5.80T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 7.90iT - 67T^{2} \) |
| 71 | \( 1 + 7.38T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 3.32iT - 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−8.883312945868821895764159176724, −8.219788943083455423172088739629, −7.73758734235120897058343180367, −6.25844191683018519418792217953, −5.68208765426612876460938689515, −4.52732390547127397458942455292, −4.19005703409137667379389461795, −3.33055123270650164106906655684, −1.53968310316074337303597112595, −0.26095901336744707410734027162,
1.74286247715672608775805469416, 2.79809343191345424888072850840, 3.10505363099405892241110038393, 4.96744341182510447124692855168, 5.85628627050096879664199605136, 6.28945583853848073447285642858, 7.17532158741040740617554851899, 7.946011075421578842712199077210, 8.643693864775722959508601700362, 9.405668339413737896496389385616
